# bibtab

bibtype: article
filename: elefunt.bib
label: Adj:2014:SRC
author: G. Adj and F. Rodriguez-Henriquez
title: Square Root Computation over Even Extension Fields
journal: j-IEEE-TRANS-COMPUT
volume: 63
number: 11
pages: 2829--2841
month: nov
monthnumber: 11
year: 2014
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2013.145
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Thu Nov 06 07:39:04 2014
bibtimestamp: 2014.11.06 07:39:04 ???
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: G. Adj and F. Rodriguez-Henriquez
rawtitle: Square Root Computation over Even Extension Fields
bibsource:
https://www.math.utah.edu/pub/tex/bib/cryptography2010.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib

keywords:
Algorithm design and analysis; Complexity theory; Computational efficiency; Computer science; Elliptic curve cryptography; Elliptic curves; even extension fields; finite extension fields; finite field arithmetic; Modular square root; number theoretical problem; number theory; square root computation; Taxonomy

entry:
@Article{Adj:2014:SRC,
  author =       "G. Adj and F. Rodriguez-Henriquez",
  title =        "Square Root Computation over Even Extension Fields",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "63",
  number =       "11",
  pages =        "2829--2841",
  month =        nov,
  year =         "2014",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2013.145",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Thu Nov 06 07:39:04 2014",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/cryptography2010.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
  keywords =     "Algorithm design and analysis; Complexity theory;
                 Computational efficiency; Computer science; Elliptic
                 curve cryptography; Elliptic curves; even extension
                 fields; finite extension fields; finite field
                 arithmetic; Modular square root; number theoretical
                 problem; number theory; square root computation;
                 Taxonomy",
}


bibtype: misc
filename: elefunt.bib
label: Anonymous:2014:CLL
author: Anonymous
title: CR-Libm --- a library of correctly rounded elementary functions in double-precision
pages: 2829--2841
year: 2014
bibdate: Sat Oct 31 07:21:21 2015
bibtimestamp: 2015.10.31 07:21:21 ???
URL: http://lipforge.ens-lyon.fr/www/crlibm/
keywords: CR-Libm; scslib (software carry save library)
acknowledgement: ack-nhfb
howpublished: Web site
rawauthor: Anonymous
rawtitle: {CR-Libm} --- a library of correctly rounded elementary functions in double-precision
abstract:
CRlibm is a free mathematical library (libm) that provides: (1) implementations of the double-precision C99 standard elementary functions; (2) correctly rounded in the four IEEE-754 rounding modes; (3) with a comprehensive proof of both the algorithms used and their implementation; (4) sufficiently efficient in average time, worst-case time, and memory consumption to replace existing libms transparently.

rawabstract:
CRlibm is a free mathematical library (libm) that provides: (1) implementations of the double-precision C99 standard elementary functions; (2) correctly rounded in the four IEEE-754 rounding modes; (3) with a comprehensive proof of both the algorithms used and their implementation; (4) sufficiently efficient in average time, worst-case time, and memory consumption to replace existing libms transparently.

entry:
@Misc{Anonymous:2014:CLL,
  author =       "Anonymous",
  title =        "{CR-Libm} --- a library of correctly rounded
                 elementary functions in double-precision",
  howpublished = "Web site",
  year =         "2014",
  bibdate =      "Sat Oct 31 07:21:21 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://lipforge.ens-lyon.fr/www/crlibm/",
  abstract =     "CRlibm is a free mathematical library (libm) that
                 provides: (1) implementations of the double-precision
                 C99 standard elementary functions; (2) correctly
                 rounded in the four IEEE-754 rounding modes; (3) with a
                 comprehensive proof of both the algorithms used and
                 their implementation; (4) sufficiently efficient in
                 average time, worst-case time, and memory consumption
                 to replace existing libms transparently.",
  acknowledgement = ack-nhfb,
  keywords =     "CR-Libm; scslib (software carry save library)",
}


bibtype: article
filename: elefunt.bib
label: Babusci:2014:SBS
author: D. Babusci and G. Dattoli and K. Gorska and K. A. Penson
title: The spherical Bessel and Struve functions and operational methods
journal: j-APPL-MATH-COMP
volume: 238
pages: 1--6
day: 1
month: jul
monthnumber: 7
year: 2014
CODEN: AMHCBQ
ISSN: 0096-3003 (print), 1873-5649 (electronic) OR 00963003 (print), 18735649 (electronic)
ISSNL: 0096-3003 OR 00963003
bibdate: Fri May 23 10:53:19 MDT 2014
bibtimestamp: 2014.05.23 10:53:19 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0096300314005086
fjournal: Applied Mathematics and Computation
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00963003/
rawauthor: D. Babusci and G. Dattoli and K. G{\'o}rska and K. A. Penson
rawtitle: The spherical {Bessel} and {Struve} functions and operational methods
bibsource:
https://www.math.utah.edu/pub/tex/bib/applmathcomput2010.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

entry:
@Article{Babusci:2014:SBS,
  author =       "D. Babusci and G. Dattoli and K. G{\'o}rska and K. A.
                 Penson",
  title =        "The spherical {Bessel} and {Struve} functions and
                 operational methods",
  journal =      j-APPL-MATH-COMP,
  volume =       "238",
  number =       "??",
  pages =        "1--6",
  day =          "1",
  month =        jul,
  year =         "2014",
  CODEN =        "AMHCBQ",
  ISSN =         "0096-3003 (print), 1873-5649 (electronic)",
  ISSN-L =       "0096-3003",
  bibdate =      "Fri May 23 10:53:19 MDT 2014",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/applmathcomput2010.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0096300314005086",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied Mathematics and Computation",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00963003/",
}


bibtype: article
filename: elefunt.bib
label: Backeljauw:2014:VES
author: Franky Backeljauw and Stefan Becuwe and Annie Cuyt and Joris Van Deun and Daniel W. Lozier
title: Validated evaluation of special mathematical functions
journal: j-SCI-COMPUT-PROGRAM
volume: 90 (part A)
pages: 2--20
day: 15
month: sep
monthnumber: 9
year: 2014
CODEN: SCPGD4
DOI: https://doi.org/10.1016/j.scico.2013.05.006
ISSN: 0167-6423 (print), 1872-7964 (electronic) OR 01676423 (print), 18727964 (electronic)
ISSNL: 0167-6423 OR 01676423
bibdate: Thu May 22 07:49:47 MDT 2014
bibtimestamp: 2014.05.22 07:49:47 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0167642313001263
fjournal: Science of Computer Programming
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/01676423/
rawauthor: Franky Backeljauw and Stefan Becuwe and Annie Cuyt and Joris {Van Deun} and Daniel W. Lozier
rawtitle: Validated evaluation of special mathematical functions
bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/scicomputprogram.bib

entry:
@Article{Backeljauw:2014:VES,
  author =       "Franky Backeljauw and Stefan Becuwe and Annie Cuyt and
                 Joris {Van Deun} and Daniel W. Lozier",
  title =        "Validated evaluation of special mathematical
                 functions",
  journal =      j-SCI-COMPUT-PROGRAM,
  volume =       "90 (part A)",
  number =       "??",
  pages =        "2--20",
  day =          "15",
  month =        sep,
  year =         "2014",
  CODEN =        "SCPGD4",
  DOI =          "https://doi.org/10.1016/j.scico.2013.05.006",
  ISSN =         "0167-6423 (print), 1872-7964 (electronic)",
  ISSN-L =       "0167-6423",
  bibdate =      "Thu May 22 07:49:47 MDT 2014",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/scicomputprogram.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0167642313001263",
  acknowledgement = ack-nhfb,
  fjournal =     "Science of Computer Programming",
  journal-URL =  "http://www.sciencedirect.com/science/journal/01676423/",
}


bibtype: book
filename: elefunt.bib
label: Bartsch:2014:TMF
author: Hans-Jochen Bartsch
publisher: Fachbuchverlag Leipzig im Hanser-Verlag
address: Munchen, Germany
edition: Twenty-third
pages: 832
year: 2014
ISBN: 3-446-43800-9 OR 3446438009
ISBN13: 978-3-446-43800-2 OR 9783446438002
bibdate: Wed Mar 1 17:30:07 MST 2017
bibsource: fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2017.03.01 17:30:07 MST
URL: http://d-nb.info/1045240478/04
language: German
acknowledgement: ack-nhfb
rawauthor: Hans-Jochen Bartsch
title:
Taschenbuch mathematischer Formeln fur Ingenieure und Naturwissenschaftler: [Fur Studium und Beruf]. (German) [Pocketbook of mathematical formulas for engineers and natural sciences: [For study and job]]

TOC:
1 Logik, Mengen, Zahlensysteme / 21 2 Arithmetik / 46 3 Gleichungen und Ungleichungen / 91 4 Elementare Geometrie / 124 5 Lineare Algebra / 168 6 Vektoren, Analytische Geometrie / 244 7 Funktionen und Kurven / 335 8 Differenzialrechnung / 421 9 Integralrechnung / 467 10 Vektoranalysis / 512 11 Differenzialgleichungen / 536 12 Reihen, F- und L-/ Transformation 13 Statistik, Stochastik / 643 14 Integraltabellen / 719

rawtitle:
{Taschenbuch mathematischer Formeln f{\"u}r Ingenieure und Naturwissenschaftler: [F{\"u}r Studium und Beruf]}. ({German}) [{Pocketbook} of mathematical formulas for engineers and natural sciences: [For study and job]]

rawTOC:
1 Logik, Mengen, Zahlensysteme / 21 \\ 2 Arithmetik / 46 \\ 3 Gleichungen und Ungleichungen / 91 \\ 4 Elementare Geometrie / 124 \\ 5 Lineare Algebra / 168 \\ 6 Vektoren, Analytische Geometrie / 244 \\ 7 Funktionen und Kurven / 335 \\ 8 Differenzialrechnung / 421 \\ 9 Integralrechnung / 467 \\ 10 Vektoranalysis / 512 \\ 11 Differenzialgleichungen / 536 \\ 12 Reihen, F- und L-/ Transformation \\ 13 Statistik, Stochastik / 643 \\ 14 Integraltabellen / 719

entry:
@Book{Bartsch:2014:TMF,
  author =       "Hans-Jochen Bartsch",
  title =        "{Taschenbuch mathematischer Formeln f{\"u}r Ingenieure
                 und Naturwissenschaftler: [F{\"u}r Studium und Beruf]}.
                 ({German}) [{Pocketbook} of mathematical formulas for
                 engineers and natural sciences: [For study and job]]",
  publisher =    "Fachbuchverlag Leipzig im Hanser-Verlag",
  address =      "M{\"u}nchen, Germany",
  edition =      "Twenty-third",
  pages =        "832",
  year =         "2014",
  ISBN =         "3-446-43800-9",
  ISBN-13 =      "978-3-446-43800-2",
  LCCN =         "????",
  bibdate =      "Wed Mar 1 17:30:07 MST 2017",
  bibsource =    "fsz3950.oclc.org:210/WorldCat;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://d-nb.info/1045240478/04",
  acknowledgement = ack-nhfb,
  language =     "German",
  tableofcontents = "1 Logik, Mengen, Zahlensysteme / 21 \\
                 2 Arithmetik / 46 \\
                 3 Gleichungen und Ungleichungen / 91 \\
                 4 Elementare Geometrie / 124 \\
                 5 Lineare Algebra / 168 \\
                 6 Vektoren, Analytische Geometrie / 244 \\
                 7 Funktionen und Kurven / 335 \\
                 8 Differenzialrechnung / 421 \\
                 9 Integralrechnung / 467 \\
                 10 Vektoranalysis / 512 \\
                 11 Differenzialgleichungen / 536 \\
                 12 Reihen, F- und L-/ Transformation \\
                 13 Statistik, Stochastik / 643 \\
                 14 Integraltabellen / 719",
}


bibtype: book
filename: elefunt.bib
label: Boyd:2014:STE
author: John P. (John Philip) Boyd
publisher: pub-SIAM
address: pub-SIAM:adr
pages: xviii + 460
year: 2014
ISBN: 1-61197-351-1 (paperback) OR 1611973511 (paperback)
ISBN13: 978-1-61197-351-8 (paperback) OR 9781611973518 (paperback)
LCCN: QA353.T7 B69 2014
bibdate: Wed Sep 23 17:10:53 MDT 2015
bibtimestamp: 2015.09.23 17:10:53 MDT
subject: Transcendental functions; Chebyshev polynomials; Transcendental numbers
acknowledgement: ack-nhfb
authordates: 1951--
rawauthor: John P. (John Philip) Boyd
title:
Solving transcendental equations: the Chebyshev polynomial proxy and other numerical rootfinders, perturbation series, and oracles

bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/numana2010.bib; z3950.loc.gov:7090/Voyager

URL:
http://www.loc.gov/catdir/enhancements/fy1503/2014017078-b.html; http://www.loc.gov/catdir/enhancements/fy1503/2014017078-d.html; http://www.loc.gov/catdir/enhancements/fy1503/2014017078-t.html

TOC:
I: Introduction and overview Introduction: Key themes in rootfinding II: the Chebyshev-Proxy rootfinder and its generalizations The Chebyshev-Proxy/Companion matrix rootfinder Adaptive Chebyshev interpolation Adaptive Fourier interpolation and rootfinding Complex zeros: Interpolation on a disk, the Delves--Lyness algorithm, and contour integrals III: Fundamentals: Iterations, bifurcation, and continuation Newton iteration and its kin Bifurcation theory Continuation in a parameter IV: Polynomials Polynomial equations and the irony of Galois Theory The Quadratic Equation Roots of a cubic polynomial Roots of a quartic polynomial V: Analytical methods Methods for explicit solutions Regular perturbation methods for roots Singular perturbation methods: fractional powers, logarithms, and exponential asymptotics VI: Classics, special functions, inverses, and oracles Classical methods for solving one equation in one unknown Special algorithms for special functions Inverse functions of one unknown Oracles: Theorems and algorithms for determining the existence, nonexistence, and number of zeros VII: Bivariate systems Two equations in two unknowns VIII: Challenges Past and future A: Companion matrices B: Chebyshev interpolation and quadrature Marching triangles D: Imbricate-Fourier series and the Poisson Summation Theorem

rawtitle:
Solving transcendental equations: the {Chebyshev} polynomial proxy and other numerical rootfinders, perturbation series, and oracles

rawTOC:
I: Introduction and overview \\ Introduction: Key themes in rootfinding \\ II: the Chebyshev-Proxy rootfinder and its generalizations \\ The Chebyshev-Proxy/Companion matrix rootfinder \\ Adaptive Chebyshev interpolation \\ Adaptive Fourier interpolation and rootfinding \\ Complex zeros: Interpolation on a disk, the Delves--Lyness algorithm, and contour integrals \\ III: Fundamentals: Iterations, bifurcation, and continuation \\ Newton iteration and its kin \\ Bifurcation theory \\ Continuation in a parameter \\ IV: Polynomials \\ Polynomial equations and the irony of Galois Theory \\ The Quadratic Equation \\ Roots of a cubic polynomial \\ Roots of a quartic polynomial \\ V: Analytical methods \\ Methods for explicit solutions \\ Regular perturbation methods for roots \\ Singular perturbation methods: fractional powers, logarithms, and exponential asymptotics \\ VI: Classics, special functions, inverses, and oracles \\ Classical methods for solving one equation in one unknown \\ Special algorithms for special functions \\ Inverse functions of one unknown \\ Oracles: Theorems and algorithms for determining the existence, nonexistence, and number of zeros \\ VII: Bivariate systems \\ Two equations in two unknowns \\ VIII: Challenges \\ Past and future \\ A: Companion matrices \\ B: Chebyshev interpolation and quadrature \\ Marching triangles \\ D: Imbricate-Fourier series and the Poisson Summation Theorem

entry:
@Book{Boyd:2014:STE,
  author =       "John P. (John Philip) Boyd",
  title =        "Solving transcendental equations: the {Chebyshev}
                 polynomial proxy and other numerical rootfinders,
                 perturbation series, and oracles",
  publisher =    pub-SIAM,
  address =      pub-SIAM:adr,
  pages =        "xviii + 460",
  year =         "2014",
  ISBN =         "1-61197-351-1 (paperback)",
  ISBN-13 =      "978-1-61197-351-8 (paperback)",
  LCCN =         "QA353.T7 B69 2014",
  bibdate =      "Wed Sep 23 17:10:53 MDT 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/numana2010.bib;
                 z3950.loc.gov:7090/Voyager",
  URL =          "http://www.loc.gov/catdir/enhancements/fy1503/2014017078-b.html;
                 http://www.loc.gov/catdir/enhancements/fy1503/2014017078-d.html;
                 http://www.loc.gov/catdir/enhancements/fy1503/2014017078-t.html",
  acknowledgement = ack-nhfb,
  author-dates = "1951--",
  subject =      "Transcendental functions; Chebyshev polynomials;
                 Transcendental numbers",
  tableofcontents = "I: Introduction and overview \\
                 Introduction: Key themes in rootfinding \\
                 II: the Chebyshev-Proxy rootfinder and its
                 generalizations \\
                 The Chebyshev-Proxy/Companion matrix rootfinder \\
                 Adaptive Chebyshev interpolation \\
                 Adaptive Fourier interpolation and rootfinding \\
                 Complex zeros: Interpolation on a disk, the
                 Delves--Lyness algorithm, and contour integrals \\
                 III: Fundamentals: Iterations, bifurcation, and
                 continuation \\
                 Newton iteration and its kin \\
                 Bifurcation theory \\
                 Continuation in a parameter \\
                 IV: Polynomials \\
                 Polynomial equations and the irony of Galois Theory \\
                 The Quadratic Equation \\
                 Roots of a cubic polynomial \\
                 Roots of a quartic polynomial \\
                 V: Analytical methods \\
                 Methods for explicit solutions \\
                 Regular perturbation methods for roots \\
                 Singular perturbation methods: fractional powers,
                 logarithms, and exponential asymptotics \\
                 VI: Classics, special functions, inverses, and oracles
                 \\
                 Classical methods for solving one equation in one
                 unknown \\
                 Special algorithms for special functions \\
                 Inverse functions of one unknown \\
                 Oracles: Theorems and algorithms for determining the
                 existence, nonexistence, and number of zeros \\
                 VII: Bivariate systems \\
                 Two equations in two unknowns \\
                 VIII: Challenges \\
                 Past and future \\
                 A: Companion matrices \\
                 B: Chebyshev interpolation and quadrature \\
                 Marching triangles \\
                 D: Imbricate-Fourier series and the Poisson Summation
                 Theorem",
}


bibtype: article
filename: elefunt.bib
label: Buehler:2014:CCH
author: Stephan Buehler and Claude Duhr
title: CHAPLIN-Complex Harmonic Polylogarithms in Fortran
journal: j-COMP-PHYS-COMM
volume: 185
number: 10
pages: 2703--2713
month: oct
monthnumber: 10
year: 2014
CODEN: CPHCBZ
ISSN: 0010-4655 (print), 1879-2944 (electronic) OR 00104655 (print), 18792944 (electronic)
ISSNL: 0010-4655 OR 00104655
bibdate: Sat Aug 16 08:37:41 MDT 2014
bibtimestamp: 2014.08.16 08:37:41 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0010465514001969
fjournal: Computer Physics Communications
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00104655/
rawauthor: Stephan Buehler and Claude Duhr
rawtitle: {CHAPLIN-Complex Harmonic Polylogarithms} in {Fortran}
bibsource:
https://www.math.utah.edu/pub/tex/bib/compphyscomm2010.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fortran3.bib

entry:
@Article{Buehler:2014:CCH,
  author =       "Stephan Buehler and Claude Duhr",
  title =        "{CHAPLIN-Complex Harmonic Polylogarithms} in
                 {Fortran}",
  journal =      j-COMP-PHYS-COMM,
  volume =       "185",
  number =       "10",
  pages =        "2703--2713",
  month =        oct,
  year =         "2014",
  CODEN =        "CPHCBZ",
  ISSN =         "0010-4655 (print), 1879-2944 (electronic)",
  ISSN-L =       "0010-4655",
  bibdate =      "Sat Aug 16 08:37:41 MDT 2014",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/compphyscomm2010.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fortran3.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0010465514001969",
  acknowledgement = ack-nhfb,
  fjournal =     "Computer Physics Communications",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00104655/",
}


bibtype: article
filename: elefunt.bib
label: Choudhury:2014:SAA
author: Amit Choudhury
title: A simple approximation to the area under standard normal curve
journal: Mathematics and Statistics
volume: 2
number: 3
pages: 147--149
year: 2014
DOI: https://doi.org/10.13189/ms.2014.020307
ISSN: 2332-2071 (print), 2332-2144 (electronic) OR 23322071 (print), 23322144 (electronic)
ISSNL: 2332-2071 OR 23322071
bibdate: Sat Dec 16 15:57:04 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2017.12.16 15:57:04 ???
acknowledgement: ack-nhfb
rawauthor: Amit Choudhury
rawtitle: A simple approximation to the area under standard normal curve
URL:
http://www.hrpub.org/journals/article_info.php?aid=1470; https://www.hrpub.org/download/20140305/MS7-13401470.pdf

entry:
@Article{Choudhury:2014:SAA,
  author =       "Amit Choudhury",
  title =        "A simple approximation to the area under standard
                 normal curve",
  journal =      "Mathematics and Statistics",
  volume =       "2",
  number =       "3",
  pages =        "147--149",
  month =        "????",
  year =         "2014",
  DOI =          "https://doi.org/10.13189/ms.2014.020307",
  ISSN =         "2332-2071 (print), 2332-2144 (electronic)",
  ISSN-L =       "2332-2071",
  bibdate =      "Sat Dec 16 15:57:04 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.hrpub.org/journals/article_info.php?aid=1470;
                 https://www.hrpub.org/download/20140305/MS7-13401470.pdf",
  acknowledgement = ack-nhfb,
}


bibtype: book
filename: elefunt.bib
label: Dunkl:2014:OPS
author: Charles F. Dunkl and Yuan Xu
title: Orthogonal Polynomials of Several Variables
volume: 155
publisher: pub-CAMBRIDGE
address: pub-CAMBRIDGE:adr
edition: Second
pages: xvii + 420
year: 2014
ISBN: 1-107-07189-5, 1-316-05717-8 (e-book) OR 1107071895, 1316057178 (ebook)
ISBN13: 978-1-107-07189-6, 978-1-316-05717-9 (e-book) OR 9781107071896, 9781316057179 (ebook)
LCCN: QA404.5 .D86 2014
bibdate: Sat Nov 11 06:43:34 MST 2023
bibsource: fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2023.11.11 06:43:34 MST
series: Encyclopedia of mathematics and its applications
remark: See also first edition cite Dunkl:2001:OPS
acknowledgement: ack-nhfb
rawauthor: Charles F. Dunkl and Yuan Xu
rawtitle: Orthogonal Polynomials of Several Variables
URL:
http://catdir.loc.gov/catdir/enhancements/fy1408/2014001846-b.html; http://catdir.loc.gov/catdir/enhancements/fy1408/2014001846-d.html; http://catdir.loc.gov/catdir/enhancements/fy1408/2014001846-t.html; http://digitale-objekte.hbz-nrw.de/storage2/2014/11/17/file_11/5877666.pdf

subject:
Orthogonal polynomials; Functions of several real variables; Polynomes orthogonaux; Fonctions de plusieurs variables reelles; Functions of several real variables; Orthogonal polynomials; Orthogonale reeksen; Ortogonalpolynom

TOC:
Preface to the Second Edition / xiii Preface to the First Edition / xv 1 Background / 1 1.1 The Gamma and Beta Functions / 1 1.2 Hypergeometric Series / 3 1.2.1 Lauricella series / 5 1.3 Orthogonal Polynomials of One Variable / 6 1.3.1 General properties / 6 1.3.2 Three-term recurrence / 9 1.4 Classical Orthogonal Polynomials / 13 1.4.1 Hermite polynomials / 13 1.4.2 Laguerre polynomials / 14 1.4.3 Gegenbauer polynomials / 16 1.4.4 Jacobi polynomials / 20 1.5 Modified Classical Polynomials / 22 1.5.1 Generalized Hermite polynomials / 24 1.5.2 Generalized Gegenbauer polynomials / 25 1.5.3 A limiting relation / 27 1.6 Notes / 27 2 Orthogonal Polynomials in Two Variables / 28 2.1 Introduction / 28 2.2 Product Orthogonal Polynomials / 29 2.3 Orthogonal Polynomials on the Unit Disk / 30 2.4 Orthogonal Polynomials on the Triangle / 35 2.5 Orthogonal Polynomials and Differential Equations / 37 2.6 Generating Orthogonal Polynomials of Two Variables / 38 2.6.1 A method for generating orthogonal polynomials / 38 2.6.2 Orthogonal polynomials for a radial weight / 40 2.6.3 Orthogonal polynomials in complex variables / 41 2.7 First Family of Koornwinder Polynomials / 45 2.8 A Related Family of Orthogonal Polynomials / 43 2.9 Second Family of Koornwinder Polynomials / 50 2.10 Notes / 54 3 General Properties of Orthogonal Polynomials in Several Variables / 57 3.1 Notation and Preliminaries / 58 3.2 Moment Functionals and Orthogonal Polynomials in Several Variables / 60 3.2.1 Definition of orthogonal polynomials / 60 3.2.2 Orthogonal polynomials and moment matrices / 64 3.2.3 The moment problem / 67 3.3 The Three-Term Relation / 70 3.3.1 Definition and basic properties / 70 3.3.2 Favard's theorem / 73 3.3.3 Centrally symmetric integrals / 76 3.3.4 Examples / 79 3.4 Jacobi Matrices and Commuting Operators / 82 3.5 Further Properties of the Three-Term Relation / 87 3.5.1 Recurrence formula / 87 3.5.2 General solutions of the three-term relation / 94 3.6 Reproducing Kernels and Fourier Orthogonal Series / 96 3.6.1 Reproducing kernels / 97 3.6.2 Fourier orthogonal series / 101 3.7 Common Zeros of Orthogonal Polynomials in Several Variables / 103 3.8 Gaussian Cubature Formulae / 107 3.9 Notes / 112 4 Orthogonal Polynomials on the Unit Sphere / 114 4.1 Spherical Harmonics / 114 4.2 Orthogonal Structures on $S^d$ and on $B^d$ / 119 4.3 Orthogonal Structures on $B^d$ and on $S^d + m - 1$ / 125 4.4 Orthogonal Structures on the Simplex / 129 4.5 Van der Corput--Schaake Inequality / 133 4.6 Notes / 136 5 Examples of Orthogonal Polynomials in Several Variables / 137 5.1 Orthogonal Polynomials for Simple Weight Functions / 137 5.1.1 Product weight functions / 138 5.1.2 Rotation-invariant weight functions / 138 5.1.3 Multiple Hermite polynomials on $mathbb R^d$ / 139 5.1.4 Multiple Laguerre polynomials on $mathbb R^d__$ / 141 5.2 Classical Orthogonal Polynomials on the Unit Ball / 141 5.2.1 Orthonormal bases / 142 5.2.2 Appell's monic orthogonal and biorthogonal polynomials / 143 5.2.3 Reproducing kernel with respect to $W_mu ^B$ on $B^d$ / 148 53.3 Classical Orthogonal Polynomials on the Simplex / 150 5.4 Orthogonal Polynomials via Symmetric Functions / 154 5.4.1 Two general families of orthogonal polynomials / 154 5.4.2 Common zeros and Gaussian cubature formulae / 156 5.5 Chebyshev Polynomials of Type $cal A_d$ / 165 5.6 Sobolev Orthogonal Polynomials on the Unit Ball / 165 5.6.1 Sobolev orthogonal polynomials defined via the gradient operator / 165 5.6.2 Sobolev orthogonal polynomials defined via the Laplacian operator / 168 5.7 Notes / 171 6 Root Systems and Coxeter Groups / 174 6.1 Introduction and Overview / 174 6.2 Root Systems / 176 6.2.1 Type $A_d - 1$ / 179 6.2.2 Type $B_d$ / 179 6.2.3 Type $I_2(m)$ / 180 6.2.4 Type $D_d$ / 181 6.2.5 Type $H_3$ / 181 6.2.6 Type $F_4$ / 182 6.2.7 Other types / 182 6.2.8 Miscellaneous results / 182 6.3 Invariant Polynomials / 183 6.3.1 Type $A_d - 1$ invariants / 183 6.3.2 Type $B_d$ invariants / 186 6.3.3 Type $D_d$ invariants / 186 6.3.4 Type $I_2(m)$ invariants / 186 6.3.5 Type $H_3$ invariants / 186 6.3.6 Type $F_4$ invariants / 187 6.4 Differential--Difference Operators / 187 6.5 The Intertwining Operator / 192 6.6 The $kappa $-Analogue of the Exponential / 200 6.7 Invariant Differential Operators / 202 6.8 Notes / 207 7 Spherical Harmonics Associated with Reflection Groups / 208 7.1 $h$-Harmonic Polynomials / 208 7.2 Inner Products on Polynomials / 217 7.3 Reproducing Kernels and the Poisson Kernel / 221 7.4 Integration of the Intertwining Operator / 224 7.5 Example: Abelian Group $cal Z_2^d$ / 228 7.5.1 Orthogonal basis for $h$-harmonics / 228 7.5.2 Intertwining and projection operators / 232 7.5.3 Monic orthogonal basis / 235 7.6 Example: Dihedral Groups / 240 7.6.1 An orthonormal basis of $cal H_(h^2_alpha , beta )$ / 241 7.6.2 Cauchy and Poisson kernels / 248 7.7 The Dunkl Transform / 250 7.8 Notes / 256 8 Generalized Classical Orthogonal Polynomials / 258 8.1 Generalized Classical Orthogonal Polynomials on the Ball / 258 8.1.1 Definition and differential-difference equations / 258 8.1.2 Orthogonal basis and reproducing kernel / 263 8.1.3 Orthogonal polynomials for $cal Z_2^d$-invariant weight functions / 266 8.1.4 Reproducing kernel for $cal Z_2^d$-invariant weight functions / 268 8.2 Generalized Classical Orthogonal Polynomials on the Simplex / 271 8.2.1 Weight function and differential-difference equation / 271 8.2.2 Orthogonal basis and reproducing kernel / 273 8.2.3 Monic orthogonal polynomials / 287 8.3 Generalized Hermite Polynomials / 278 8.4 Generalized Laguerre Polynomials / 283 8.5 Notes / 287 9 Summability of Orthogonal Expansions / 289 9.1 General Results on Orthogonal Expansions / 289 9.1.1 Uniform convergence of partial sums / 289 9.1.2 Cesaro means of the orthogonal expansion / 293 9.2 Orthogonal Expansion on the Sphere / 296 9.3 Orthogonal Expansion on the Ball / 299 9.4 Orthogonal Expansion on the Simplex / 304 9.5 Orthogonal Expansion of Laguerre and Hermite Polynomials / 306 9.6 Multiple Jacobi Expansion / 311 9.7 Notes / 315 10 Orthogonal Polynomials Associated with Symmetric Groups / 318 10.1 Partitions, Compositions and Orderings / 318 10.2 Commuting Self-Adjoint Operators / 320 10.3 The Dual Polynomial Basis / 322 10.4 $S_d$-Invariant Subspaces / 329 10.5 Degree-Changing Recurrences / 334 10.6 Norm Formulae / 337 10.6.1 Hook-length products and the pairing norm / 337 10.6.2 The biorthogonal-type norm / 341 10.6.3 The torus inner product / 343 10.6.4 Monic polynomials / 346 10.6.5 Normalizing constants / 346 10.7 Symmetric Functions and Jack Polynomials / 350 10.8 Miscellaneous Topics / 357 10.9 Notes / 362 11 Orthogonal Polynomials Associated with Octahedral Groups and Applications / 364 11.1 Introduction / 364 11.2 Operators of Type $B$ / 365 11.3 Polynomial Eigenfunctions of Type $B$ / 368 11.4 Generalized Binomial Coefficients / 376 11.5 Hermite Polynomials of Type $B$ / 373 11.6 Calogero--Sutherland Systems / 385 11.6.1 The simple harmonic oscillator / 386 11.6.2 Root systems and the Laplacian / 387 11.6.3 Type $A$ models on the line / 387 11.6.4 Type $A$ models on the circle / 389 11.6.5 Type $B$ models on the line / 392 11.7 Notes / 394 References / 396 Author Index / 413 Symbol Index / 416 Subject Index / 418

shortTOC:
Preface to the Second Edition / xiii Preface to the First Edition / xv 1 Background / 1 2 Orthogonal Polynomials in Two Variables / 28 3 General Properties of Orthogonal Polynomials in Several Variables / 57 4 Orthogonal Polynomials on the Unit Sphere / 114 5 Examples of Orthogonal Polynomials in Several Variables / 137 6 Root Systems and Coxeter Groups / 174 7 Spherical Harmonics Associated with Reflection Groups / 208 8 Generalized Classical Orthogonal Polynomials / 258 9 Summability of Orthogonal Expansions / 289 10 Orthogonal Polynomials Associated with Symmetric Groups / 318 11 Orthogonal Polynomials Associated with Octahedral Groups and Applications / 364 References / 396 Author Index / 413 Symbol Index / 416 Subject Index / 418

rawshortTOC:
Preface to the Second Edition / xiii \\ Preface to the First Edition / xv \\ 1 Background / 1 \\ 2 Orthogonal Polynomials in Two Variables / 28 \\ 3 General Properties of Orthogonal Polynomials in Several Variables / 57 \\ 4 Orthogonal Polynomials on the Unit Sphere / 114 \\ 5 Examples of Orthogonal Polynomials in Several Variables / 137 \\ 6 Root Systems and Coxeter Groups / 174 \\ 7 Spherical Harmonics Associated with Reflection Groups / 208 \\ 8 Generalized Classical Orthogonal Polynomials / 258 \\ 9 Summability of Orthogonal Expansions / 289 \\ 10 Orthogonal Polynomials Associated with Symmetric Groups / 318 \\ 11 Orthogonal Polynomials Associated with Octahedral Groups and Applications / 364 \\ References / 396 \\ Author Index / 413 \\ Symbol Index / 416 \\ Subject Index / 418

rawTOC:
Preface to the Second Edition / xiii \\ Preface to the First Edition / xv \\ \\ 1 Background / 1 \\ 1.1 The Gamma and Beta Functions / 1 \\ 1.2 Hypergeometric Series / 3 \\ 1.2.1 Lauricella series / 5 \\ 1.3 Orthogonal Polynomials of One Variable / 6 \\ 1.3.1 General properties / 6 \\ 1.3.2 Three-term recurrence / 9 \\ 1.4 Classical Orthogonal Polynomials / 13 \\ 1.4.1 Hermite polynomials / 13 \\ 1.4.2 Laguerre polynomials / 14 \\ 1.4.3 Gegenbauer polynomials / 16 \\ 1.4.4 Jacobi polynomials / 20 \\ 1.5 Modified Classical Polynomials / 22 \\ 1.5.1 Generalized Hermite polynomials / 24 \\ 1.5.2 Generalized Gegenbauer polynomials / 25 \\ 1.5.3 A limiting relation / 27 \\ 1.6 Notes / 27 \\ \\ 2 Orthogonal Polynomials in Two Variables / 28 \\ 2.1 Introduction / 28 \\ 2.2 Product Orthogonal Polynomials / 29 \\ 2.3 Orthogonal Polynomials on the Unit Disk / 30 \\ 2.4 Orthogonal Polynomials on the Triangle / 35 \\ 2.5 Orthogonal Polynomials and Differential Equations / 37 \\ 2.6 Generating Orthogonal Polynomials of Two Variables / 38 \\ 2.6.1 A method for generating orthogonal polynomials / 38 \\ 2.6.2 Orthogonal polynomials for a radial weight / 40 \\ 2.6.3 Orthogonal polynomials in complex variables / 41 \\ 2.7 First Family of Koornwinder Polynomials / 45 \\ 2.8 A Related Family of Orthogonal Polynomials / 43 \\ 2.9 Second Family of Koornwinder Polynomials / 50 \\ 2.10 Notes / 54 \\ \\ 3 General Properties of Orthogonal Polynomials in Several Variables / 57 \\ 3.1 Notation and Preliminaries / 58 \\ 3.2 Moment Functionals and Orthogonal Polynomials in Several Variables / 60 \\ 3.2.1 Definition of orthogonal polynomials / 60 \\ 3.2.2 Orthogonal polynomials and moment matrices / 64 \\ 3.2.3 The moment problem / 67 \\ 3.3 The Three-Term Relation / 70 \\ 3.3.1 Definition and basic properties / 70 \\ 3.3.2 Favard's theorem / 73 \\ 3.3.3 Centrally symmetric integrals / 76 \\ 3.3.4 Examples / 79 \\ 3.4 Jacobi Matrices and Commuting Operators / 82 \\ 3.5 Further Properties of the Three-Term Relation / 87 \\ 3.5.1 Recurrence formula / 87 \\ 3.5.2 General solutions of the three-term relation / 94 \\ 3.6 Reproducing Kernels and Fourier Orthogonal Series / 96 \\ 3.6.1 Reproducing kernels / 97 \\ 3.6.2 Fourier orthogonal series / 101 \\ 3.7 Common Zeros of Orthogonal Polynomials in Several Variables / 103 \\ 3.8 Gaussian Cubature Formulae / 107 \\ 3.9 Notes / 112 \\ \\ 4 Orthogonal Polynomials on the Unit Sphere / 114 \\ 4.1 Spherical Harmonics / 114 \\ 4.2 Orthogonal Structures on $S^d$ and on $B^d$ / 119 \\ 4.3 Orthogonal Structures on $B^d$ and on $S^{d + m - 1}$ / 125 \\ 4.4 Orthogonal Structures on the Simplex / 129 \\ 4.5 Van der Corput--Schaake Inequality / 133 \\ 4.6 Notes / 136 \\ \\ 5 Examples of Orthogonal Polynomials in Several Variables / 137 \\ 5.1 Orthogonal Polynomials for Simple Weight Functions / 137 \\ 5.1.1 Product weight functions / 138 \\ 5.1.2 Rotation-invariant weight functions / 138 \\ 5.1.3 Multiple Hermite polynomials on $\mathbb{R}^d$ / 139 \\ 5.1.4 Multiple Laguerre polynomials on $\mathbb{R}^d__$ / 141 \\ 5.2 Classical Orthogonal Polynomials on the Unit Ball / 141 \\ 5.2.1 Orthonormal bases / 142 \\ 5.2.2 Appell's monic orthogonal and biorthogonal polynomials / 143 \\ 5.2.3 Reproducing kernel with respect to $W_\mu^B$ on $B^d$ / 148 \\ 53.3 Classical Orthogonal Polynomials on the Simplex / 150 \\ 5.4 Orthogonal Polynomials via Symmetric Functions / 154 \\ 5.4.1 Two general families of orthogonal polynomials / 154 \\ 5.4.2 Common zeros and Gaussian cubature formulae / 156 \\ 5.5 Chebyshev Polynomials of Type ${\cal A}_d$ / 165 \\ 5.6 Sobolev Orthogonal Polynomials on the Unit Ball / 165 \\ 5.6.1 Sobolev orthogonal polynomials defined via the gradient operator / 165 \\ 5.6.2 Sobolev orthogonal polynomials defined via the Laplacian operator / 168 \\ 5.7 Notes / 171 \\ \\ 6 Root Systems and Coxeter Groups / 174 \\ 6.1 Introduction and Overview / 174 \\ 6.2 Root Systems / 176 \\ 6.2.1 Type $A_{d - 1}$ / 179 \\ 6.2.2 Type $B_d$ / 179 \\ 6.2.3 Type $I_2(m)$ / 180 \\ 6.2.4 Type $D_d$ / 181 \\ 6.2.5 Type $H_3$ / 181 \\ 6.2.6 Type $F_4$ / 182 \\ 6.2.7 Other types / 182 \\ 6.2.8 Miscellaneous results / 182 \\ 6.3 Invariant Polynomials / 183 \\ 6.3.1 Type $A_{d - 1}$ invariants / 183 \\ 6.3.2 Type $B_d$ invariants / 186 \\ 6.3.3 Type $D_d$ invariants / 186 \\ 6.3.4 Type $I_2(m)$ invariants / 186 \\ 6.3.5 Type $H_3$ invariants / 186 \\ 6.3.6 Type $F_4$ invariants / 187 \\ 6.4 Differential--Difference Operators / 187 \\ 6.5 The Intertwining Operator / 192 \\ 6.6 The $\kappa$-Analogue of the Exponential / 200 \\ 6.7 Invariant Differential Operators / 202 \\ 6.8 Notes / 207 \\ \\ 7 Spherical Harmonics Associated with Reflection Groups / 208 \\ 7.1 $h$-Harmonic Polynomials / 208 \\ 7.2 Inner Products on Polynomials / 217 \\ 7.3 Reproducing Kernels and the Poisson Kernel / 221 \\ 7.4 Integration of the Intertwining Operator / 224 \\ 7.5 Example: Abelian Group ${\cal Z}_2^d$ / 228 \\ 7.5.1 Orthogonal basis for $h$-harmonics / 228 \\ 7.5.2 Intertwining and projection operators / 232 \\ 7.5.3 Monic orthogonal basis / 235 \\ 7.6 Example: Dihedral Groups / 240 \\ 7.6.1 An orthonormal basis of ${\cal H}_(h^2_{\alpha, \beta})$ / 241 \\ 7.6.2 Cauchy and Poisson kernels / 248 \\ 7.7 The Dunkl Transform / 250 \\ 7.8 Notes / 256 \\ \\ 8 Generalized Classical Orthogonal Polynomials / 258 \\ 8.1 Generalized Classical Orthogonal Polynomials on the Ball / 258 \\ 8.1.1 Definition and differential-difference equations / 258 \\ 8.1.2 Orthogonal basis and reproducing kernel / 263 \\ 8.1.3 Orthogonal polynomials for ${\cal Z}_2^d$-invariant weight functions / 266 \\ 8.1.4 Reproducing kernel for ${\cal Z}_2^d$-invariant weight functions / 268 \\ 8.2 Generalized Classical Orthogonal Polynomials on the Simplex / 271 \\ 8.2.1 Weight function and differential-difference equation / 271 \\ 8.2.2 Orthogonal basis and reproducing kernel / 273 \\ 8.2.3 Monic orthogonal polynomials / 287 \\ 8.3 Generalized Hermite Polynomials / 278 \\ 8.4 Generalized Laguerre Polynomials / 283 \\ 8.5 Notes / 287 \\ \\ 9 Summability of Orthogonal Expansions / 289 \\ 9.1 General Results on Orthogonal Expansions / 289 \\ 9.1.1 Uniform convergence of partial sums / 289 \\ 9.1.2 Ces{\`a}ro means of the orthogonal expansion / 293 \\ 9.2 Orthogonal Expansion on the Sphere / 296 \\ 9.3 Orthogonal Expansion on the Ball / 299 \\ 9.4 Orthogonal Expansion on the Simplex / 304 \\ 9.5 Orthogonal Expansion of Laguerre and Hermite Polynomials / 306 \\ 9.6 Multiple Jacobi Expansion / 311 \\ 9.7 Notes / 315 \\ \\ 10 Orthogonal Polynomials Associated with Symmetric Groups / 318 \\ 10.1 Partitions, Compositions and Orderings / 318 \\ 10.2 Commuting Self-Adjoint Operators / 320 \\ 10.3 The Dual Polynomial Basis / 322 \\ 10.4 $S_d$-Invariant Subspaces / 329 \\ 10.5 Degree-Changing Recurrences / 334 \\ 10.6 Norm Formulae / 337 \\ 10.6.1 Hook-length products and the pairing norm / 337 \\ 10.6.2 The biorthogonal-type norm / 341 \\ 10.6.3 The torus inner product / 343 \\ 10.6.4 Monic polynomials / 346 \\ 10.6.5 Normalizing constants / 346 \\ 10.7 Symmetric Functions and Jack Polynomials / 350 \\ 10.8 Miscellaneous Topics / 357 \\ 10.9 Notes / 362 \\ \\ 11 Orthogonal Polynomials Associated with Octahedral Groups and Applications / 364 \\ 11.1 Introduction / 364 \\ 11.2 Operators of Type $B$ / 365 \\ 11.3 Polynomial Eigenfunctions of Type $B$ / 368 \\ 11.4 Generalized Binomial Coefficients / 376 \\ 11.5 Hermite Polynomials of Type $B$ / 373 \\ 11.6 Calogero--Sutherland Systems / 385 \\ 11.6.1 The simple harmonic oscillator / 386 \\ 11.6.2 Root systems and the Laplacian / 387 \\ 11.6.3 Type $A$ models on the line / 387 \\ 11.6.4 Type $A$ models on the circle / 389 \\ 11.6.5 Type $B$ models on the line / 392 \\ 11.7 Notes / 394 \\ \\ References / 396 \\ Author Index / 413 \\ Symbol Index / 416 \\ Subject Index / 418

entry:
@Book{Dunkl:2014:OPS,
  author =       "Charles F. Dunkl and Yuan Xu",
  title =        "Orthogonal Polynomials of Several Variables",
  volume =       "155",
  publisher =    pub-CAMBRIDGE,
  address =      pub-CAMBRIDGE:adr,
  edition =      "Second",
  pages =        "xvii + 420",
  year =         "2014",
  ISBN =         "1-107-07189-5, 1-316-05717-8 (e-book)",
  ISBN-13 =      "978-1-107-07189-6, 978-1-316-05717-9 (e-book)",
  LCCN =         "QA404.5 .D86 2014",
  bibdate =      "Sat Nov 11 06:43:34 MST 2023",
  bibsource =    "fsz3950.oclc.org:210/WorldCat;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  series =       "Encyclopedia of mathematics and its applications",
  URL =          "http://catdir.loc.gov/catdir/enhancements/fy1408/2014001846-b.html;
                 http://catdir.loc.gov/catdir/enhancements/fy1408/2014001846-d.html;
                 http://catdir.loc.gov/catdir/enhancements/fy1408/2014001846-t.html;
                 http://digitale-objekte.hbz-nrw.de/storage2/2014/11/17/file_11/5877666.pdf",
  acknowledgement = ack-nhfb,
  remark =       "See also first edition \cite{Dunkl:2001:OPS}",
  shorttableofcontents = "Preface to the Second Edition / xiii \\
                 Preface to the First Edition / xv \\
                 1 Background / 1 \\
                 2 Orthogonal Polynomials in Two Variables / 28 \\
                 3 General Properties of Orthogonal Polynomials in
                 Several Variables / 57 \\
                 4 Orthogonal Polynomials on the Unit Sphere / 114 \\
                 5 Examples of Orthogonal Polynomials in Several
                 Variables / 137 \\
                 6 Root Systems and Coxeter Groups / 174 \\
                 7 Spherical Harmonics Associated with Reflection Groups
                 / 208 \\
                 8 Generalized Classical Orthogonal Polynomials / 258
                 \\
                 9 Summability of Orthogonal Expansions / 289 \\
                 10 Orthogonal Polynomials Associated with Symmetric
                 Groups / 318 \\
                 11 Orthogonal Polynomials Associated with Octahedral
                 Groups and Applications / 364 \\
                 References / 396 \\
                 Author Index / 413 \\
                 Symbol Index / 416 \\
                 Subject Index / 418",
  subject =      "Orthogonal polynomials; Functions of several real
                 variables; Polyn{\^o}mes orthogonaux; Fonctions de
                 plusieurs variables r{\'e}elles; Functions of several
                 real variables; Orthogonal polynomials; Orthogonale
                 reeksen; Ortogonalpolynom",
  tableofcontents = "Preface to the Second Edition / xiii \\
                 Preface to the First Edition / xv \\
                 \\
                 1 Background / 1 \\
                 1.1 The Gamma and Beta Functions / 1 \\
                 1.2 Hypergeometric Series / 3 \\
                 1.2.1 Lauricella series / 5 \\
                 1.3 Orthogonal Polynomials of One Variable / 6 \\
                 1.3.1 General properties / 6 \\
                 1.3.2 Three-term recurrence / 9 \\
                 1.4 Classical Orthogonal Polynomials / 13 \\
                 1.4.1 Hermite polynomials / 13 \\
                 1.4.2 Laguerre polynomials / 14 \\
                 1.4.3 Gegenbauer polynomials / 16 \\
                 1.4.4 Jacobi polynomials / 20 \\
                 1.5 Modified Classical Polynomials / 22 \\
                 1.5.1 Generalized Hermite polynomials / 24 \\
                 1.5.2 Generalized Gegenbauer polynomials / 25 \\
                 1.5.3 A limiting relation / 27 \\
                 1.6 Notes / 27 \\
                 \\
                 2 Orthogonal Polynomials in Two Variables / 28 \\
                 2.1 Introduction / 28 \\
                 2.2 Product Orthogonal Polynomials / 29 \\
                 2.3 Orthogonal Polynomials on the Unit Disk / 30 \\
                 2.4 Orthogonal Polynomials on the Triangle / 35 \\
                 2.5 Orthogonal Polynomials and Differential Equations /
                 37 \\
                 2.6 Generating Orthogonal Polynomials of Two Variables
                 / 38 \\
                 2.6.1 A method for generating orthogonal polynomials /
                 38 \\
                 2.6.2 Orthogonal polynomials for a radial weight / 40
                 \\
                 2.6.3 Orthogonal polynomials in complex variables / 41
                 \\
                 2.7 First Family of Koornwinder Polynomials / 45 \\
                 2.8 A Related Family of Orthogonal Polynomials / 43 \\
                 2.9 Second Family of Koornwinder Polynomials / 50 \\
                 2.10 Notes / 54 \\
                 \\
                 3 General Properties of Orthogonal Polynomials in
                 Several Variables / 57 \\
                 3.1 Notation and Preliminaries / 58 \\
                 3.2 Moment Functionals and Orthogonal Polynomials in
                 Several Variables / 60 \\
                 3.2.1 Definition of orthogonal polynomials / 60 \\
                 3.2.2 Orthogonal polynomials and moment matrices / 64
                 \\
                 3.2.3 The moment problem / 67 \\
                 3.3 The Three-Term Relation / 70 \\
                 3.3.1 Definition and basic properties / 70 \\
                 3.3.2 Favard's theorem / 73 \\
                 3.3.3 Centrally symmetric integrals / 76 \\
                 3.3.4 Examples / 79 \\
                 3.4 Jacobi Matrices and Commuting Operators / 82 \\
                 3.5 Further Properties of the Three-Term Relation / 87
                 \\
                 3.5.1 Recurrence formula / 87 \\
                 3.5.2 General solutions of the three-term relation / 94
                 \\
                 3.6 Reproducing Kernels and Fourier Orthogonal Series /
                 96 \\
                 3.6.1 Reproducing kernels / 97 \\
                 3.6.2 Fourier orthogonal series / 101 \\
                 3.7 Common Zeros of Orthogonal Polynomials in Several
                 Variables / 103 \\
                 3.8 Gaussian Cubature Formulae / 107 \\
                 3.9 Notes / 112 \\
                 \\
                 4 Orthogonal Polynomials on the Unit Sphere / 114 \\
                 4.1 Spherical Harmonics / 114 \\
                 4.2 Orthogonal Structures on $S^d$ and on $B^d$ / 119
                 \\
                 4.3 Orthogonal Structures on $B^d$ and on $S^{d + m -
                 1}$ / 125 \\
                 4.4 Orthogonal Structures on the Simplex / 129 \\
                 4.5 Van der Corput--Schaake Inequality / 133 \\
                 4.6 Notes / 136 \\
                 \\
                 5 Examples of Orthogonal Polynomials in Several
                 Variables / 137 \\
                 5.1 Orthogonal Polynomials for Simple Weight Functions
                 / 137 \\
                 5.1.1 Product weight functions / 138 \\
                 5.1.2 Rotation-invariant weight functions / 138 \\
                 5.1.3 Multiple Hermite polynomials on $\mathbb{R}^d$ /
                 139 \\
                 5.1.4 Multiple Laguerre polynomials on $\mathbb{R}^d__$
                 / 141 \\
                 5.2 Classical Orthogonal Polynomials on the Unit Ball /
                 141 \\
                 5.2.1 Orthonormal bases / 142 \\
                 5.2.2 Appell's monic orthogonal and biorthogonal
                 polynomials / 143 \\
                 5.2.3 Reproducing kernel with respect to $W_\mu^B$ on
                 $B^d$ / 148 \\
                 53.3 Classical Orthogonal Polynomials on the Simplex /
                 150 \\
                 5.4 Orthogonal Polynomials via Symmetric Functions /
                 154 \\
                 5.4.1 Two general families of orthogonal polynomials /
                 154 \\
                 5.4.2 Common zeros and Gaussian cubature formulae / 156
                 \\
                 5.5 Chebyshev Polynomials of Type ${\cal A}_d$ / 165
                 \\
                 5.6 Sobolev Orthogonal Polynomials on the Unit Ball /
                 165 \\
                 5.6.1 Sobolev orthogonal polynomials defined via the
                 gradient operator / 165 \\
                 5.6.2 Sobolev orthogonal polynomials defined via the
                 Laplacian operator / 168 \\
                 5.7 Notes / 171 \\
                 \\
                 6 Root Systems and Coxeter Groups / 174 \\
                 6.1 Introduction and Overview / 174 \\
                 6.2 Root Systems / 176 \\
                 6.2.1 Type $A_{d - 1}$ / 179 \\
                 6.2.2 Type $B_d$ / 179 \\
                 6.2.3 Type $I_2(m)$ / 180 \\
                 6.2.4 Type $D_d$ / 181 \\
                 6.2.5 Type $H_3$ / 181 \\
                 6.2.6 Type $F_4$ / 182 \\
                 6.2.7 Other types / 182 \\
                 6.2.8 Miscellaneous results / 182 \\
                 6.3 Invariant Polynomials / 183 \\
                 6.3.1 Type $A_{d - 1}$ invariants / 183 \\
                 6.3.2 Type $B_d$ invariants / 186 \\
                 6.3.3 Type $D_d$ invariants / 186 \\
                 6.3.4 Type $I_2(m)$ invariants / 186 \\
                 6.3.5 Type $H_3$ invariants / 186 \\
                 6.3.6 Type $F_4$ invariants / 187 \\
                 6.4 Differential--Difference Operators / 187 \\
                 6.5 The Intertwining Operator / 192 \\
                 6.6 The $\kappa$-Analogue of the Exponential / 200 \\
                 6.7 Invariant Differential Operators / 202 \\
                 6.8 Notes / 207 \\
                 \\
                 7 Spherical Harmonics Associated with Reflection Groups
                 / 208 \\
                 7.1 $h$-Harmonic Polynomials / 208 \\
                 7.2 Inner Products on Polynomials / 217 \\
                 7.3 Reproducing Kernels and the Poisson Kernel / 221
                 \\
                 7.4 Integration of the Intertwining Operator / 224 \\
                 7.5 Example: Abelian Group ${\cal Z}_2^d$ / 228 \\
                 7.5.1 Orthogonal basis for $h$-harmonics / 228 \\
                 7.5.2 Intertwining and projection operators / 232 \\
                 7.5.3 Monic orthogonal basis / 235 \\
                 7.6 Example: Dihedral Groups / 240 \\
                 7.6.1 An orthonormal basis of ${\cal H}_(h^2_{\alpha,
                 \beta})$ / 241 \\
                 7.6.2 Cauchy and Poisson kernels / 248 \\
                 7.7 The Dunkl Transform / 250 \\
                 7.8 Notes / 256 \\
                 \\
                 8 Generalized Classical Orthogonal Polynomials / 258
                 \\
                 8.1 Generalized Classical Orthogonal Polynomials on the
                 Ball / 258 \\
                 8.1.1 Definition and differential-difference equations
                 / 258 \\
                 8.1.2 Orthogonal basis and reproducing kernel / 263 \\
                 8.1.3 Orthogonal polynomials for ${\cal
                 Z}_2^d$-invariant weight functions / 266 \\
                 8.1.4 Reproducing kernel for ${\cal Z}_2^d$-invariant
                 weight functions / 268 \\
                 8.2 Generalized Classical Orthogonal Polynomials on the
                 Simplex / 271 \\
                 8.2.1 Weight function and differential-difference
                 equation / 271 \\
                 8.2.2 Orthogonal basis and reproducing kernel / 273 \\
                 8.2.3 Monic orthogonal polynomials / 287 \\
                 8.3 Generalized Hermite Polynomials / 278 \\
                 8.4 Generalized Laguerre Polynomials / 283 \\
                 8.5 Notes / 287 \\
                 \\
                 9 Summability of Orthogonal Expansions / 289 \\
                 9.1 General Results on Orthogonal Expansions / 289 \\
                 9.1.1 Uniform convergence of partial sums / 289 \\
                 9.1.2 Ces{\`a}ro means of the orthogonal expansion /
                 293 \\
                 9.2 Orthogonal Expansion on the Sphere / 296 \\
                 9.3 Orthogonal Expansion on the Ball / 299 \\
                 9.4 Orthogonal Expansion on the Simplex / 304 \\
                 9.5 Orthogonal Expansion of Laguerre and Hermite
                 Polynomials / 306 \\
                 9.6 Multiple Jacobi Expansion / 311 \\
                 9.7 Notes / 315 \\
                 \\
                 10 Orthogonal Polynomials Associated with Symmetric
                 Groups / 318 \\
                 10.1 Partitions, Compositions and Orderings / 318 \\
                 10.2 Commuting Self-Adjoint Operators / 320 \\
                 10.3 The Dual Polynomial Basis / 322 \\
                 10.4 $S_d$-Invariant Subspaces / 329 \\
                 10.5 Degree-Changing Recurrences / 334 \\
                 10.6 Norm Formulae / 337 \\
                 10.6.1 Hook-length products and the pairing norm / 337
                 \\
                 10.6.2 The biorthogonal-type norm / 341 \\
                 10.6.3 The torus inner product / 343 \\
                 10.6.4 Monic polynomials / 346 \\
                 10.6.5 Normalizing constants / 346 \\
                 10.7 Symmetric Functions and Jack Polynomials / 350 \\
                 10.8 Miscellaneous Topics / 357 \\
                 10.9 Notes / 362 \\
                 \\
                 11 Orthogonal Polynomials Associated with Octahedral
                 Groups and Applications / 364 \\
                 11.1 Introduction / 364 \\
                 11.2 Operators of Type $B$ / 365 \\
                 11.3 Polynomial Eigenfunctions of Type $B$ / 368 \\
                 11.4 Generalized Binomial Coefficients / 376 \\
                 11.5 Hermite Polynomials of Type $B$ / 373 \\
                 11.6 Calogero--Sutherland Systems / 385 \\
                 11.6.1 The simple harmonic oscillator / 386 \\
                 11.6.2 Root systems and the Laplacian / 387 \\
                 11.6.3 Type $A$ models on the line / 387 \\
                 11.6.4 Type $A$ models on the circle / 389 \\
                 11.6.5 Type $B$ models on the line / 392 \\
                 11.7 Notes / 394 \\
                 \\
                 References / 396 \\
                 Author Index / 413 \\
                 Symbol Index / 416 \\
                 Subject Index / 418",
}


bibtype: article
filename: elefunt.bib
label: Fukushima:2014:ACG
author: Toshio Fukushima
title: Analytical computation of generalized Fermi--Dirac integrals by truncated Sommerfeld expansions
journal: j-APPL-MATH-COMP
volume: 234
pages: 417--433
day: 15
month: may
monthnumber: 5
year: 2014
CODEN: AMHCBQ
ISSN: 0096-3003 (print), 1873-5649 (electronic) OR 00963003 (print), 18735649 (electronic)
ISSNL: 0096-3003 OR 00963003
bibdate: Mon Apr 21 18:04:13 MDT 2014
bibtimestamp: 2014.04.21 18:04:13 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0096300314002926
fjournal: Applied Mathematics and Computation
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00963003/
rawauthor: Toshio Fukushima
rawtitle: Analytical computation of generalized {Fermi--Dirac} integrals by truncated {Sommerfeld} expansions
entry:
@Article{Fukushima:2014:ACG,
  author =       "Toshio Fukushima",
  title =        "Analytical computation of generalized {Fermi--Dirac}
                 integrals by truncated {Sommerfeld} expansions",
  journal =      j-APPL-MATH-COMP,
  volume =       "234",
  number =       "??",
  pages =        "417--433",
  day =          "15",
  month =        may,
  year =         "2014",
  CODEN =        "AMHCBQ",
  ISSN =         "0096-3003 (print), 1873-5649 (electronic)",
  ISSN-L =       "0096-3003",
  bibdate =      "Mon Apr 21 18:04:13 MDT 2014",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/applmathcomput2010.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0096300314002926",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied Mathematics and Computation",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00963003/",
}


bibtype: article
filename: elefunt.bib
label: Fukushima:2014:CGI
author: Toshio Fukushima
title: Computation of a general integral of Fermi--Dirac distribution by McDougall--Stoner method
journal: j-APPL-MATH-COMP
volume: 238
pages: 485--510
day: 1
month: jul
monthnumber: 7
year: 2014
CODEN: AMHCBQ
ISSN: 0096-3003 (print), 1873-5649 (electronic) OR 00963003 (print), 18735649 (electronic)
ISSNL: 0096-3003 OR 00963003
bibdate: Fri May 23 10:53:19 MDT 2014
bibtimestamp: 2014.05.23 10:53:19 MDT
URL: http://www.sciencedirect.com/science/article/pii/S009630031400561X
fjournal: Applied Mathematics and Computation
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00963003/
rawauthor: Toshio Fukushima
rawtitle: Computation of a general integral of {Fermi--Dirac} distribution by {McDougall--Stoner} method
bibsource:
https://www.math.utah.edu/pub/bibnet/authors/d/dirac-p-a-m.bib; https://www.math.utah.edu/pub/bibnet/authors/f/fermi-enrico.bib; https://www.math.utah.edu/pub/tex/bib/applmathcomput2010.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

entry:
@Article{Fukushima:2014:CGI,
  author =       "Toshio Fukushima",
  title =        "Computation of a general integral of {Fermi--Dirac}
                 distribution by {McDougall--Stoner} method",
  journal =      j-APPL-MATH-COMP,
  volume =       "238",
  number =       "??",
  pages =        "485--510",
  day =          "1",
  month =        jul,
  year =         "2014",
  CODEN =        "AMHCBQ",
  ISSN =         "0096-3003 (print), 1873-5649 (electronic)",
  ISSN-L =       "0096-3003",
  bibdate =      "Fri May 23 10:53:19 MDT 2014",
  bibsource =    "https://www.math.utah.edu/pub/bibnet/authors/d/dirac-p-a-m.bib;
                 https://www.math.utah.edu/pub/bibnet/authors/f/fermi-enrico.bib;
                 https://www.math.utah.edu/pub/tex/bib/applmathcomput2010.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S009630031400561X",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied Mathematics and Computation",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00963003/",
}


bibtype: article
filename: elefunt.bib
label: Gil:2014:ACM
author: Amparo Gil and Javier Segura and Nico M. Temme
title: Algorithm 939: Computation of the Marcum $Q$-Function
journal: j-TOMS
volume: 40
number: 3
pages: 20:1--20:21
month: apr
monthnumber: 4
year: 2014
CODEN: ACMSCU
DOI: https://doi.org/10.1145/2591004
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
bibdate: Mon Apr 21 17:42:14 MDT 2014
bibtimestamp: 2014.04.21 17:42:14 MDT
fjournal: ACM Transactions on Mathematical Software (TOMS)
acknowledgement: ack-nhfb
articleno: 20
journalURL: https://dl.acm.org/loi/toms
rawauthor: Amparo Gil and Javier Segura and Nico M. Temme
rawtitle: {Algorithm 939}: Computation of the {Marcum} {$Q$}-Function
abstract:
Methods and an algorithm for computing the generalized Marcum $Q$-function $ (Q_mu (x, y))$ and the complementary function $ (P_mu (x, y))$ are described. These functions appear in problems of different technical and scientific areas such as, for example, radar detection and communications, statistics, and probability theory, where they are called the noncentral chi-square or the noncentral gamma cumulative distribution functions. The algorithm for computing the Marcum functions combines different methods of evaluation in different regions: series expansions, integral representations, asymptotic expansions, and use of three-term homogeneous recurrence relations. A relative accuracy close to $ 10^-12$ can be obtained in the parameter region $ (x, y, mu ) in [0, A] times [0, A] times [1, A]$, $ A = 200$, while for larger parameters the accuracy decreases (close to $ 10^-11$ for $ A = 1000$ and close to $ 5 times 10^-11$ for $ A = 10000$).

rawabstract:
Methods and an algorithm for computing the generalized Marcum $Q$-function $ (Q_\mu (x, y))$ and the complementary function $ (P_\mu (x, y))$ are described. These functions appear in problems of different technical and scientific areas such as, for example, radar detection and communications, statistics, and probability theory, where they are called the noncentral chi-square or the noncentral gamma cumulative distribution functions. The algorithm for computing the Marcum functions combines different methods of evaluation in different regions: series expansions, integral representations, asymptotic expansions, and use of three-term homogeneous recurrence relations. A relative accuracy close to $ 10^{-12}$ can be obtained in the parameter region $ (x, y, \mu) \in [0, A] \times [0, A] \times [1, A]$, $ A = 200$, while for larger parameters the accuracy decreases (close to $ 10^{-11}$ for $ A = 1000$ and close to $ 5 \times 10^{-11}$ for $ A = 10000$).

entry:
@Article{Gil:2014:ACM,
  author =       "Amparo Gil and Javier Segura and Nico M. Temme",
  title =        "{Algorithm 939}: Computation of the {Marcum}
                 {$Q$}-Function",
  journal =      j-TOMS,
  volume =       "40",
  number =       "3",
  pages =        "20:1--20:21",
  month =        apr,
  year =         "2014",
  CODEN =        "ACMSCU",
  DOI =          "https://doi.org/10.1145/2591004",
  ISSN =         "0098-3500 (print), 1557-7295 (electronic)",
  ISSN-L =       "0098-3500",
  bibdate =      "Mon Apr 21 17:42:14 MDT 2014",
  bibsource =    "http://www.acm.org/pubs/contents/journals/toms/;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/toms.bib",
  abstract =     "Methods and an algorithm for computing the generalized
                 Marcum $Q$-function $ (Q_\mu (x, y))$ and the
                 complementary function $ (P_\mu (x, y))$ are described.
                 These functions appear in problems of different
                 technical and scientific areas such as, for example,
                 radar detection and communications, statistics, and
                 probability theory, where they are called the
                 noncentral chi-square or the noncentral gamma
                 cumulative distribution functions. The algorithm for
                 computing the Marcum functions combines different
                 methods of evaluation in different regions: series
                 expansions, integral representations, asymptotic
                 expansions, and use of three-term homogeneous
                 recurrence relations. A relative accuracy close to $
                 10^{-12}$ can be obtained in the parameter region $ (x,
                 y, \mu) \in [0, A] \times [0, A] \times [1, A]$, $ A =
                 200$, while for larger parameters the accuracy
                 decreases (close to $ 10^{-11}$ for $ A = 1000$ and
                 close to $ 5 \times 10^{-11}$ for $ A = 10000$).",
  acknowledgement = ack-nhfb,
  articleno =    "20",
  fjournal =     "ACM Transactions on Mathematical Software (TOMS)",
  journal-URL =  "https://dl.acm.org/loi/toms",
}


bibtype: article
filename: elefunt.bib
label: Gil:2014:CZA
author: Amparo Gil and Javier Segura
title: On the complex zeros of Airy and Bessel functions and those of their derivatives
journal: j-ANAL-APPL
volume: 12
number: 5
pages: 537--561
month: aug
monthnumber: 8
year: 2014
DOI: https://doi.org/10.1142/s0219530514500341
ISSN: 0219-5305 (print), 1793-6861 (electronic) OR 02195305 (print), 17936861 (electronic)
ISSNL: 0219-5305 OR 02195305
bibdate: Thu Nov 16 07:32:34 2023
bibtimestamp: 2023.11.16 07:32:34 ???
note: Special Issue: Dedicated to the Memory of Frank Olver (Part II).
fjournal: Analysis and Applications (Singapore)
acknowledgement: ack-nhfb
ajournal: Anal. Appl. (Singapore)
journalURL: https://www.worldscientific.com/worldscinet/aa
subjectdates: Frank William John Olver (15 December 1924--23 April 2013)
rawauthor: Amparo Gil and Javier Segura
rawnote: Special Issue: Dedicated to the Memory of Frank Olver (Part II).
rawtitle: On the complex zeros of {Airy} and {Bessel} functions and those of their derivatives
bibsource:
https://www.math.utah.edu/pub/bibnet/authors/o/olver-frank-w-j.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

entry:
@Article{Gil:2014:CZA,
  author =       "Amparo Gil and Javier Segura",
  title =        "On the complex zeros of {Airy} and {Bessel} functions
                 and those of their derivatives",
  journal =      j-ANAL-APPL,
  volume =       "12",
  number =       "5",
  pages =        "537--561",
  month =        aug,
  year =         "2014",
  DOI =          "https://doi.org/10.1142/s0219530514500341",
  ISSN =         "0219-5305 (print), 1793-6861 (electronic)",
  ISSN-L =       "0219-5305",
  bibdate =      "Thu Nov 16 07:32:34 2023",
  bibsource =    "https://www.math.utah.edu/pub/bibnet/authors/o/olver-frank-w-j.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  note =         "Special Issue: Dedicated to the Memory of Frank Olver
                 (Part II).",
  acknowledgement = ack-nhfb,
  ajournal =     "Anal. Appl. (Singapore)",
  fjournal =     "Analysis and Applications (Singapore)",
  journal-URL =  "https://www.worldscientific.com/worldscinet/aa",
  subject-dates = "Frank William John Olver (15 December 1924--23 April
                 2013)",
}


bibtype: article
filename: elefunt.bib
label: Gil:2014:RSD
author: Amparo Gil and Javier Segura and Nico M. Temme
title: Recent software developments for special functions in the Santander--Amsterdam project
journal: j-SCI-COMPUT-PROGRAM
volume: 90 (part A)
pages: 42--54
day: 15
month: sep
monthnumber: 9
year: 2014
CODEN: SCPGD4
DOI: https://doi.org/10.1016/j.scico.2013.11.004
ISSN: 0167-6423 (print), 1872-7964 (electronic) OR 01676423 (print), 18727964 (electronic)
ISSNL: 0167-6423 OR 01676423
bibdate: Thu May 22 07:49:47 MDT 2014
bibtimestamp: 2014.05.22 07:49:47 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0167642313002888
fjournal: Science of Computer Programming
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/01676423/
rawauthor: Amparo Gil and Javier Segura and Nico M. Temme
rawtitle: Recent software developments for special functions in the {Santander--Amsterdam} project
bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/scicomputprogram.bib

entry:
@Article{Gil:2014:RSD,
  author =       "Amparo Gil and Javier Segura and Nico M. Temme",
  title =        "Recent software developments for special functions in
                 the {Santander--Amsterdam} project",
  journal =      j-SCI-COMPUT-PROGRAM,
  volume =       "90 (part A)",
  number =       "??",
  pages =        "42--54",
  day =          "15",
  month =        sep,
  year =         "2014",
  CODEN =        "SCPGD4",
  DOI =          "https://doi.org/10.1016/j.scico.2013.11.004",
  ISSN =         "0167-6423 (print), 1872-7964 (electronic)",
  ISSN-L =       "0167-6423",
  bibdate =      "Thu May 22 07:49:47 MDT 2014",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/scicomputprogram.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0167642313002888",
  acknowledgement = ack-nhfb,
  fjournal =     "Science of Computer Programming",
  journal-URL =  "http://www.sciencedirect.com/science/journal/01676423/",
}


bibtype: article
filename: elefunt.bib
label: Goerg:2014:ULW
author: Georg M. Goerg
title: Usage of the Lambert $W$ function in statistics
journal: j-ANN-APPL-STAT
volume: 8
number: 4
pages: 2567--2567
month: dec
monthnumber: 12
year: 2014
ISSN: 1932-6157 (print), 1941-7330 (electronic) OR 19326157 (print), 19417330 (electronic)
ISSNL: 1932-6157 OR 19326157
bibdate: Wed Feb 11 19:26:08 MST 2015
bibtimestamp: 2015.02.11 19:26:08 MST
URL: http://projecteuclid.org/euclid.aoas/1419001755
fjournal: Annals of Applied Statistics
acknowledgement: ack-nhfb
journalURL: http://projecteuclid.org/all/euclid.aoas/; http://www.jstor.org/journals/19326157.html
rawauthor: Georg M. Goerg
rawtitle: Usage of the {Lambert} {$W$} function in statistics
bibsource:
https://www.math.utah.edu/pub/tex/bib/annapplstat.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

entry:
@Article{Goerg:2014:ULW,
  author =       "Georg M. Goerg",
  title =        "Usage of the {Lambert} {$W$} function in statistics",
  journal =      j-ANN-APPL-STAT,
  volume =       "8",
  number =       "4",
  pages =        "2567--2567",
  month =        dec,
  year =         "2014",
  CODEN =        "????",
  ISSN =         "1932-6157 (print), 1941-7330 (electronic)",
  ISSN-L =       "1932-6157",
  bibdate =      "Wed Feb 11 19:26:08 MST 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/annapplstat.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://projecteuclid.org/euclid.aoas/1419001755",
  acknowledgement = ack-nhfb,
  fjournal =     "Annals of Applied Statistics",
  journal-URL =  "http://projecteuclid.org/all/euclid.aoas/;
                 http://www.jstor.org/journals/19326157.html",
}


bibtype: inproceedings
filename: elefunt.bib
label: Greuel:2014:SIS
author: Gert-Martin Greuel and Wolfram Sperber
title: swMATH --- an Information Service for Mathematical Software
crossref: Hong:2014:MSI
pages: 691--701
year: 2014
DOI: https://doi.org/10.1007/978-3-662-44199-2_103
bibdate: Tue Sep 26 10:21:48 2023
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2023.09.26 10:21:48 ???
acknowledgement: ack-nhfb
rawauthor: Gert-Martin Greuel and Wolfram Sperber
rawtitle: {swMATH} --- an Information Service for Mathematical Software
entry:
@InProceedings{Greuel:2014:SIS,
  author =       "Gert-Martin Greuel and Wolfram Sperber",
  title =        "{swMATH} --- an Information Service for Mathematical
                 Software",
  crossref =     "Hong:2014:MSI",
  pages =        "691--701",
  year =         "2014",
  DOI =          "https://doi.org/10.1007/978-3-662-44199-2_103",
  bibdate =      "Tue Sep 26 10:21:48 2023",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: elefunt.bib
label: Greynat:2014:NAE
author: David Greynat and Javier Sesma
title: A new approach to the epsilon expansion of generalized hypergeometric functions
journal: j-COMP-PHYS-COMM
volume: 185
number: 2
pages: 472--478
month: feb
monthnumber: 2
year: 2014
CODEN: CPHCBZ
ISSN: 0010-4655 (print), 1879-2944 (electronic) OR 00104655 (print), 18792944 (electronic)
ISSNL: 0010-4655 OR 00104655
bibdate: Mon Dec 2 12:05:01 MST 2013
bibtimestamp: 2013.12.02 12:05:01 MST
URL: http://www.sciencedirect.com/science/article/pii/S001046551300324X
fjournal: Computer Physics Communications
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00104655
rawauthor: David Greynat and Javier Sesma
rawtitle: A new approach to the epsilon expansion of generalized hypergeometric functions
entry:
@Article{Greynat:2014:NAE,
  author =       "David Greynat and Javier Sesma",
  title =        "A new approach to the epsilon expansion of generalized
                 hypergeometric functions",
  journal =      j-COMP-PHYS-COMM,
  volume =       "185",
  number =       "2",
  pages =        "472--478",
  month =        feb,
  year =         "2014",
  CODEN =        "CPHCBZ",
  ISSN =         "0010-4655 (print), 1879-2944 (electronic)",
  ISSN-L =       "0010-4655",
  bibdate =      "Mon Dec 2 12:05:01 MST 2013",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/compphyscomm2010.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S001046551300324X",
  acknowledgement = ack-nhfb,
  fjournal =     "Computer Physics Communications",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00104655",
}


bibtype: article
filename: elefunt.bib
label: Harvey:2014:SAC
author: David Harvey
title: A subquadratic algorithm for computing the $n$-th Bernoulli number
journal: j-MATH-COMPUT
volume: 83
number: 289
pages: 2471--2477
year: 2014
CODEN: MCMPAF
ISSN: 0025-5718 (print), 1088-6842 (electronic) OR 00255718 (print), 10886842 (electronic)
ISSNL: 0025-5718 OR 00255718
bibdate: Tue Sep 9 11:37:57 MDT 2014
bibtimestamp: 2014.09.09 11:37:57 MDT
fjournal: Mathematics of Computation
acknowledgement: ack-nhfb
journalURL: http://www.ams.org/mcom/
rawauthor: David Harvey
rawtitle: A subquadratic algorithm for computing the $n$-th {Bernoulli} number
bibsource:
http://www.ams.org/mcom/2014-83-289; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp2010.bib

URL:
http://www.ams.org/journals/mcom/2014-83-289/S0025-5718-2014-02832-9; http://www.ams.org/journals/mcom/2014-83-289/S0025-5718-2014-02832-9/S0025-5718-2014-02832-9.pdf

entry:
@Article{Harvey:2014:SAC,
  author =       "David Harvey",
  title =        "A subquadratic algorithm for computing the $n$-th
                 {Bernoulli} number",
  journal =      j-MATH-COMPUT,
  volume =       "83",
  number =       "289",
  pages =        "2471--2477",
  year =         "2014",
  CODEN =        "MCMPAF",
  ISSN =         "0025-5718 (print), 1088-6842 (electronic)",
  ISSN-L =       "0025-5718",
  bibdate =      "Tue Sep 9 11:37:57 MDT 2014",
  bibsource =    "http://www.ams.org/mcom/2014-83-289;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/mathcomp2010.bib",
  URL =          "http://www.ams.org/journals/mcom/2014-83-289/S0025-5718-2014-02832-9;
                 http://www.ams.org/journals/mcom/2014-83-289/S0025-5718-2014-02832-9/S0025-5718-2014-02832-9.pdf",
  acknowledgement = ack-nhfb,
  fjournal =     "Mathematics of Computation",
  journal-URL =  "http://www.ams.org/mcom/",
}


bibtype: article
filename: elefunt.bib
label: Johansson:2014:EIE
author: Fredrik Johansson
title: Efficient implementation of elementary functions in the medium-precision range
journal: arxiv.org
pages: ??--??
day: 27
month: oct
monthnumber: 10
year: 2014
bibdate: Mon Jun 12 16:12:02 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2017.06.12 16:12:02 ???
URL: https://arxiv.org/abs/1410.7176
acknowledgement: ack-nhfb
rawauthor: Fredrik Johansson
rawtitle: Efficient implementation of elementary functions in the medium-precision range
abstract:
We describe a new implementation of the elementary transcendental functions exp, sin, cos, log and atan for variable precision up to approximately 4096 bits. Compared to the MPFR library, we achieve a maximum speedup ranging from a factor 3 for cos to 30 for atan. Our implementation uses table-based argument reduction together with rectangular splitting to evaluate Taylor series. We collect denominators to reduce the number of divisions in the Taylor series, and avoid overhead by doing all multiprecision arithmetic using the mpn layer of the GMP library. Our implementation provides rigorous error bounds.

rawabstract:
We describe a new implementation of the elementary transcendental functions exp, sin, cos, log and atan for variable precision up to approximately 4096 bits. Compared to the MPFR library, we achieve a maximum speedup ranging from a factor 3 for cos to 30 for atan. Our implementation uses table-based argument reduction together with rectangular splitting to evaluate Taylor series. We collect denominators to reduce the number of divisions in the Taylor series, and avoid overhead by doing all multiprecision arithmetic using the mpn layer of the GMP library. Our implementation provides rigorous error bounds.

entry:
@Article{Johansson:2014:EIE,
  author =       "Fredrik Johansson",
  title =        "Efficient implementation of elementary functions in
                 the medium-precision range",
  journal =      "arxiv.org",
  volume =       "??",
  number =       "??",
  pages =        "??--??",
  day =          "27",
  month =        oct,
  year =         "2014",
  bibdate =      "Mon Jun 12 16:12:02 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "https://arxiv.org/abs/1410.7176",
  abstract =     "We describe a new implementation of the elementary
                 transcendental functions exp, sin, cos, log and atan
                 for variable precision up to approximately 4096 bits.
                 Compared to the MPFR library, we achieve a maximum
                 speedup ranging from a factor 3 for cos to 30 for atan.
                 Our implementation uses table-based argument reduction
                 together with rectangular splitting to evaluate Taylor
                 series. We collect denominators to reduce the number of
                 divisions in the Taylor series, and avoid overhead by
                 doing all multiprecision arithmetic using the mpn layer
                 of the GMP library. Our implementation provides
                 rigorous error bounds.",
  acknowledgement = ack-nhfb,
}


bibtype: phdthesis
filename: elefunt.bib
label: Johansson:2014:FRC
author: Fredrik Johansson
title: Fast and Rigorous Computation of Special Functions to High Precision
type: Ph.D. thesis
school: Johannes Kepler University
address: Linz, Austria
pages: ix + 109
day: 24
month: mar
monthnumber: 3
year: 2014
bibdate: Sat Aug 09 09:01:13 2014
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2014.08.09 09:01:13 ???
URL: http://permalink.obvsg.at/AC10776210
remark: Reviewed in booktitle ACM Communications in Computer Algebra, bf 48(2) 28--28 (2014).
acknowledgement: ack-nhfb
rawauthor: Fredrik Johansson
rawtitle: Fast and Rigorous Computation of Special Functions to High Precision
abstract:
The problem of efficiently evaluating special functions to high precision has been considered by numerous authors. Important tools used for this purpose include algorithms for evaluation of linearly recurrent sequences, and algorithms for power series arithmetic.par In this work, we give new baby-step, giant-step algorithms for evaluation of linearly recurrent sequences involving an expensive parameter (such as a high-precision real number) and for computing compositional inverses of power series. Our algorithms do not have the best asymptotic complexity, but they are faster than previous algorithms in practice over a large input range.par Using a combination of techniques, we also obtain efficient new algorithms for numerically evaluating the gamma function $ Gamma (z) $ and the Hurwitz zeta function $ zeta (s, a) $, or Taylor series expansions of those functions, with rigorous error bounds. Our methods achieve softly optimal complexity when computing a large number of derivatives to proportionally high precision.par Finally, we show that isolated values of the integer partition function $ p(n) $ can be computed rigorously with softly optimal complexity by means of the Hardy--Ramanujan--Rademacher formula and careful numerical evaluation. We provide open source implementations which run significantly faster than previously published software. The implementations are used for record computations of the partition function, including the tabulation of several billion Ramanujan-type congruences, and of Taylor series associated with the Riemann zeta function.

rawabstract:
The problem of efficiently evaluating special functions to high precision has been considered by numerous authors. Important tools used for this purpose include algorithms for evaluation of linearly recurrent sequences, and algorithms for power series arithmetic.\par In this work, we give new baby-step, giant-step algorithms for evaluation of linearly recurrent sequences involving an expensive parameter (such as a high-precision real number) and for computing compositional inverses of power series. Our algorithms do not have the best asymptotic complexity, but they are faster than previous algorithms in practice over a large input range.\par Using a combination of techniques, we also obtain efficient new algorithms for numerically evaluating the gamma function $ \Gamma (z) $ and the Hurwitz zeta function $ \zeta (s, a) $, or Taylor series expansions of those functions, with rigorous error bounds. Our methods achieve softly optimal complexity when computing a large number of derivatives to proportionally high precision.\par Finally, we show that isolated values of the integer partition function $ p(n) $ can be computed rigorously with softly optimal complexity by means of the Hardy--Ramanujan--Rademacher formula and careful numerical evaluation. We provide open source implementations which run significantly faster than previously published software. The implementations are used for record computations of the partition function, including the tabulation of several billion Ramanujan-type congruences, and of Taylor series associated with the Riemann zeta function.

entry:
@PhdThesis{Johansson:2014:FRC,
  author =       "Fredrik Johansson",
  title =        "Fast and Rigorous Computation of Special Functions to
                 High Precision",
  type =         "{Ph.D.} thesis",
  school =       "Johannes Kepler University",
  address =      "Linz, Austria",
  pages =        "ix + 109",
  day =          "24",
  month =        mar,
  year =         "2014",
  bibdate =      "Sat Aug 09 09:01:13 2014",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://permalink.obvsg.at/AC10776210",
  abstract =     "The problem of efficiently evaluating special
                 functions to high precision has been considered by
                 numerous authors. Important tools used for this purpose
                 include algorithms for evaluation of linearly recurrent
                 sequences, and algorithms for power series
                 arithmetic.\par

                 In this work, we give new baby-step, giant-step
                 algorithms for evaluation of linearly recurrent
                 sequences involving an expensive parameter (such as a
                 high-precision real number) and for computing
                 compositional inverses of power series. Our algorithms
                 do not have the best asymptotic complexity, but they
                 are faster than previous algorithms in practice over a
                 large input range.\par

                 Using a combination of techniques, we also obtain
                 efficient new algorithms for numerically evaluating the
                 gamma function $ \Gamma (z) $ and the Hurwitz zeta
                 function $ \zeta (s, a) $, or Taylor series expansions
                 of those functions, with rigorous error bounds. Our
                 methods achieve softly optimal complexity when
                 computing a large number of derivatives to
                 proportionally high precision.\par

                 Finally, we show that isolated values of the integer
                 partition function $ p(n) $ can be computed rigorously
                 with softly optimal complexity by means of the
                 Hardy--Ramanujan--Rademacher formula and careful
                 numerical evaluation. We provide open source
                 implementations which run significantly faster than
                 previously published software. The implementations are
                 used for record computations of the partition function,
                 including the tabulation of several billion
                 Ramanujan-type congruences, and of Taylor series
                 associated with the Riemann zeta function.",
  acknowledgement = ack-nhfb,
  remark =       "Reviewed in \booktitle{ACM Communications in Computer
                 Algebra}, {\bf 48}(2) 28--28 (2014).",
}


bibtype: article
filename: elefunt.bib
label: Krasikov:2014:ABA
author: Ilia Krasikov
title: Approximations for the Bessel and Airy functions with an explicit error term
journal: j-LMS-J-COMPUT-MATH
volume: 17
number: 1
pages: 209--225
year: 2014
DOI: https://doi.org/10.1112/S1461157013000351
ISSN: 1461-1570 OR 14611570
bibdate: Tue Sep 9 12:34:08 MDT 2014
bibtimestamp: 2014.09.09 12:34:08 MDT
fjournal: LMS Journal of Computation and Mathematics
acknowledgement: ack-nhfb
ajournal: LMS J. Comput. Math.
journalURL: http://journals.cambridge.org/action/displayJournal?jid=JCM
onlinedate: 19 May 2014
rawauthor: Ilia Krasikov
rawtitle: Approximations for the {Bessel} and {Airy} functions with an explicit error term
bibsource:
http://journals.cambridge.org/action/displayJournal?jid=JCM; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/lms-j-comput-math.bib

entry:
@Article{Krasikov:2014:ABA,
  author =       "Ilia Krasikov",
  title =        "Approximations for the {Bessel} and {Airy} functions
                 with an explicit error term",
  journal =      j-LMS-J-COMPUT-MATH,
  volume =       "17",
  number =       "1",
  pages =        "209--225",
  year =         "2014",
  CODEN =        "????",
  DOI =          "https://doi.org/10.1112/S1461157013000351",
  ISSN =         "1461-1570",
  bibdate =      "Tue Sep 9 12:34:08 MDT 2014",
  bibsource =    "http://journals.cambridge.org/action/displayJournal?jid=JCM;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/lms-j-comput-math.bib",
  acknowledgement = ack-nhfb,
  ajournal =     "LMS J. Comput. Math.",
  fjournal =     "LMS Journal of Computation and Mathematics",
  journal-URL =  "http://journals.cambridge.org/action/displayJournal?jid=JCM",
  onlinedate =   "19 May 2014",
}


bibtype: inproceedings
filename: elefunt.bib
label: Kupriianova:2014:MMF
author: Olga Kupriianova and Christoph Lauter
title: Metalibm: A Mathematical Functions Code Generator
crossref: Hong:2014:MSI
pages: 713--717
year: 2014
DOI: https://doi.org/10.1007/978-3-662-44199-2_106
bibdate: Tue Sep 26 10:21:48 2023
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2023.09.26 10:21:48 ???
acknowledgement: ack-nhfb
rawauthor: Olga Kupriianova and Christoph Lauter
rawtitle: {Metalibm}: A Mathematical Functions Code Generator
entry:
@InProceedings{Kupriianova:2014:MMF,
  author =       "Olga Kupriianova and Christoph Lauter",
  title =        "{Metalibm}: A Mathematical Functions Code Generator",
  crossref =     "Hong:2014:MSI",
  pages =        "713--717",
  year =         "2014",
  DOI =          "https://doi.org/10.1007/978-3-662-44199-2_106",
  bibdate =      "Tue Sep 26 10:21:48 2023",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: elefunt.bib
label: Li:2014:ICH
author: Dingfang Li and Ping Liu and Jisheng Kou
title: An improvement of Chebyshev--Halley methods free from second derivative
journal: j-APPL-MATH-COMP
volume: 235
pages: 221--225
day: 25
month: may
monthnumber: 5
year: 2014
CODEN: AMHCBQ
ISSN: 0096-3003 (print), 1873-5649 (electronic) OR 00963003 (print), 18735649 (electronic)
ISSNL: 0096-3003 OR 00963003
bibdate: Mon Apr 21 18:04:20 MDT 2014
bibtimestamp: 2014.04.21 18:04:20 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0096300314003312
fjournal: Applied Mathematics and Computation
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00963003/
rawauthor: Dingfang Li and Ping Liu and Jisheng Kou
rawtitle: An improvement of {Chebyshev--Halley} methods free from second derivative
entry:
@Article{Li:2014:ICH,
  author =       "Dingfang Li and Ping Liu and Jisheng Kou",
  title =        "An improvement of {Chebyshev--Halley} methods free
                 from second derivative",
  journal =      j-APPL-MATH-COMP,
  volume =       "235",
  number =       "??",
  pages =        "221--225",
  day =          "25",
  month =        may,
  year =         "2014",
  CODEN =        "AMHCBQ",
  ISSN =         "0096-3003 (print), 1873-5649 (electronic)",
  ISSN-L =       "0096-3003",
  bibdate =      "Mon Apr 21 18:04:20 MDT 2014",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/applmathcomput2010.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0096300314003312",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied Mathematics and Computation",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00963003/",
}


bibtype: article
filename: elefunt.bib
label: Lu:2014:GAF
author: Dawei Lu and Jinghai Feng and Congxu Ma
title: A general asymptotic formula of the gamma function based on the Burnside's formula
journal: j-J-NUMBER-THEORY
volume: 145
pages: 317--328
month: dec
monthnumber: 12
year: 2014
CODEN: JNUTA9
DOI: https://doi.org/10.1016/j.jnt.2014.06.016
ISSN: 0022-314X (print), 1096-1658 (electronic) OR 0022314X (print), 10961658 (electronic)
ISSNL: 0022-314X OR 0022314X
bibdate: Wed Jul 15 08:49:09 MDT 2020
bibtimestamp: 2020.07.15 08:49:09 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0022314X14002224
fjournal: Journal of Number Theory
acknowledgement: ack-nhfb
ajournal: J. Number Theory
journalURL: http://www.sciencedirect.com/science/journal/0022314X
rawauthor: Dawei Lu and Jinghai Feng and Congxu Ma
rawtitle: A general asymptotic formula of the gamma function based on the {Burnside}'s formula
bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jnumbertheory2010.bib

entry:
@Article{Lu:2014:GAF,
  author =       "Dawei Lu and Jinghai Feng and Congxu Ma",
  title =        "A general asymptotic formula of the gamma function
                 based on the {Burnside}'s formula",
  journal =      j-J-NUMBER-THEORY,
  volume =       "145",
  number =       "??",
  pages =        "317--328",
  month =        dec,
  year =         "2014",
  CODEN =        "JNUTA9",
  DOI =          "https://doi.org/10.1016/j.jnt.2014.06.016",
  ISSN =         "0022-314X (print), 1096-1658 (electronic)",
  ISSN-L =       "0022-314X",
  bibdate =      "Wed Jul 15 08:49:09 MDT 2020",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jnumbertheory2010.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0022314X14002224",
  acknowledgement = ack-nhfb,
  ajournal =     "J. Number Theory",
  fjournal =     "Journal of Number Theory",
  journal-URL =  "http://www.sciencedirect.com/science/journal/0022314X",
}


bibtype: article
filename: elefunt.bib
label: Lu:2014:GAG
author: Dawei Lu and Lixin Song and Congxu Ma
title: A generated approximation of the gamma function related to Windschitl's formula
journal: j-J-NUMBER-THEORY
volume: 140
pages: 215--225
month: jul
monthnumber: 7
year: 2014
CODEN: JNUTA9
DOI: https://doi.org/10.1016/j.jnt.2014.01.023
ISSN: 0022-314X (print), 1096-1658 (electronic) OR 0022314X (print), 10961658 (electronic)
ISSNL: 0022-314X OR 0022314X
bibdate: Wed Jul 15 08:49:07 MDT 2020
bibtimestamp: 2020.07.15 08:49:07 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0022314X14000687
fjournal: Journal of Number Theory
acknowledgement: ack-nhfb
ajournal: J. Number Theory
journalURL: http://www.sciencedirect.com/science/journal/0022314X
rawauthor: Dawei Lu and Lixin Song and Congxu Ma
rawtitle: A generated approximation of the gamma function related to {Windschitl}'s formula
bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jnumbertheory2010.bib

entry:
@Article{Lu:2014:GAG,
  author =       "Dawei Lu and Lixin Song and Congxu Ma",
  title =        "A generated approximation of the gamma function
                 related to {Windschitl}'s formula",
  journal =      j-J-NUMBER-THEORY,
  volume =       "140",
  number =       "??",
  pages =        "215--225",
  month =        jul,
  year =         "2014",
  CODEN =        "JNUTA9",
  DOI =          "https://doi.org/10.1016/j.jnt.2014.01.023",
  ISSN =         "0022-314X (print), 1096-1658 (electronic)",
  ISSN-L =       "0022-314X",
  bibdate =      "Wed Jul 15 08:49:07 MDT 2020",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jnumbertheory2010.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0022314X14000687",
  acknowledgement = ack-nhfb,
  ajournal =     "J. Number Theory",
  fjournal =     "Journal of Number Theory",
  journal-URL =  "http://www.sciencedirect.com/science/journal/0022314X",
}


bibtype: article
filename: elefunt.bib
label: Lu:2014:NAE
author: Dawei Lu and Xiaoguang Wang
title: A new asymptotic expansion and some inequalities for the gamma function
journal: j-J-NUMBER-THEORY
volume: 140
pages: 314--323
month: jul
monthnumber: 7
year: 2014
CODEN: JNUTA9
DOI: https://doi.org/10.1016/j.jnt.2014.01.025
ISSN: 0022-314X (print), 1096-1658 (electronic) OR 0022314X (print), 10961658 (electronic)
ISSNL: 0022-314X OR 0022314X
bibdate: Wed Jul 15 08:49:07 MDT 2020
bibtimestamp: 2020.07.15 08:49:07 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0022314X14000705
fjournal: Journal of Number Theory
acknowledgement: ack-nhfb
ajournal: J. Number Theory
journalURL: http://www.sciencedirect.com/science/journal/0022314X
rawauthor: Dawei Lu and Xiaoguang Wang
rawtitle: A new asymptotic expansion and some inequalities for the gamma function
bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jnumbertheory2010.bib

entry:
@Article{Lu:2014:NAE,
  author =       "Dawei Lu and Xiaoguang Wang",
  title =        "A new asymptotic expansion and some inequalities for
                 the gamma function",
  journal =      j-J-NUMBER-THEORY,
  volume =       "140",
  number =       "??",
  pages =        "314--323",
  month =        jul,
  year =         "2014",
  CODEN =        "JNUTA9",
  DOI =          "https://doi.org/10.1016/j.jnt.2014.01.025",
  ISSN =         "0022-314X (print), 1096-1658 (electronic)",
  ISSN-L =       "0022-314X",
  bibdate =      "Wed Jul 15 08:49:07 MDT 2020",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jnumbertheory2010.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0022314X14000705",
  acknowledgement = ack-nhfb,
  ajournal =     "J. Number Theory",
  fjournal =     "Journal of Number Theory",
  journal-URL =  "http://www.sciencedirect.com/science/journal/0022314X",
}


bibtype: article
filename: elefunt.bib
label: Lu:2014:SNI
author: Dawei Lu
title: Some new improved classes of convergence towards Euler's constant
journal: j-APPL-MATH-COMP
volume: 243
pages: 24--32
day: 15
month: sep
monthnumber: 9
year: 2014
CODEN: AMHCBQ
DOI: https://doi.org/10.1016/j.amc.2014.05.098
ISSN: 0096-3003 (print), 1873-5649 (electronic) OR 00963003 (print), 18735649 (electronic)
ISSNL: 0096-3003 OR 00963003
bibdate: Sat Aug 16 10:10:22 MDT 2014
bibtimestamp: 2014.08.16 10:10:22 MDT
URL: http://www.sciencedirect.com/science/article/pii/S009630031400798X
fjournal: Applied Mathematics and Computation
keywords: Continued fraction; Euler's constant; Inequalities; Rate of convergence
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00963003/
rawauthor: Dawei Lu
rawtitle: Some new improved classes of convergence towards {Euler}'s constant
entry:
@Article{Lu:2014:SNI,
  author =       "Dawei Lu",
  title =        "Some new improved classes of convergence towards
                 {Euler}'s constant",
  journal =      j-APPL-MATH-COMP,
  volume =       "243",
  number =       "??",
  pages =        "24--32",
  day =          "15",
  month =        sep,
  year =         "2014",
  CODEN =        "AMHCBQ",
  DOI =          "https://doi.org/10.1016/j.amc.2014.05.098",
  ISSN =         "0096-3003 (print), 1873-5649 (electronic)",
  ISSN-L =       "0096-3003",
  bibdate =      "Sat Aug 16 10:10:22 MDT 2014",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/applmathcomput2010.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S009630031400798X",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied Mathematics and Computation",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00963003/",
  keywords =     "Continued fraction; Euler's constant; Inequalities;
                 Rate of convergence",
}


bibtype: article
filename: elefunt.bib
label: Mortici:2014:SBG
author: Cristinel Mortici
title: Sharp bounds for gamma function in terms of $ x^x - 1 $
journal: j-APPL-MATH-COMP
volume: 249
pages: 278--285
day: 15
month: dec
monthnumber: 12
year: 2014
CODEN: AMHCBQ
ISSN: 0096-3003 (print), 1873-5649 (electronic) OR 00963003 (print), 18735649 (electronic)
ISSNL: 0096-3003 OR 00963003
bibdate: Wed Nov 26 10:49:00 MST 2014
bibtimestamp: 2014.11.26 10:49:00 MST
URL: http://www.sciencedirect.com/science/article/pii/S0096300314013939
fjournal: Applied Mathematics and Computation
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00963003/
rawauthor: Cristinel Mortici
rawtitle: Sharp bounds for gamma function in terms of $ x^{x - 1} $
entry:
@Article{Mortici:2014:SBG,
  author =       "Cristinel Mortici",
  title =        "Sharp bounds for gamma function in terms of $ x^{x -
                 1} $",
  journal =      j-APPL-MATH-COMP,
  volume =       "249",
  number =       "??",
  pages =        "278--285",
  day =          "15",
  month =        dec,
  year =         "2014",
  CODEN =        "AMHCBQ",
  ISSN =         "0096-3003 (print), 1873-5649 (electronic)",
  ISSN-L =       "0096-3003",
  bibdate =      "Wed Nov 26 10:49:00 MST 2014",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/applmathcomput2010.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0096300314013939",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied Mathematics and Computation",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00963003/",
}


bibtype: article
filename: elefunt.bib
label: Ogburn:2014:FDC
author: Daniel X. Ogburn and Colin L. Waters and Murray D. Sciffer and Jeff A. Hogan and Paul C. Abbott
title: A finite difference construction of the spheroidal wave functions
journal: j-COMP-PHYS-COMM
volume: 185
number: 1
pages: 244--253
month: jan
monthnumber: 1
year: 2014
CODEN: CPHCBZ
ISSN: 0010-4655 (print), 1879-2944 (electronic) OR 00104655 (print), 18792944 (electronic)
ISSNL: 0010-4655 OR 00104655
bibdate: Mon Dec 2 12:04:56 MST 2013
bibtimestamp: 2013.12.02 12:04:56 MST
URL: http://www.sciencedirect.com/science/article/pii/S0010465513002610
fjournal: Computer Physics Communications
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00104655
rawauthor: Daniel X. Ogburn and Colin L. Waters and Murray D. Sciffer and Jeff A. Hogan and Paul C. Abbott
rawtitle: A finite difference construction of the spheroidal wave functions
entry:
@Article{Ogburn:2014:FDC,
  author =       "Daniel X. Ogburn and Colin L. Waters and Murray D.
                 Sciffer and Jeff A. Hogan and Paul C. Abbott",
  title =        "A finite difference construction of the spheroidal
                 wave functions",
  journal =      j-COMP-PHYS-COMM,
  volume =       "185",
  number =       "1",
  pages =        "244--253",
  month =        jan,
  year =         "2014",
  CODEN =        "CPHCBZ",
  ISSN =         "0010-4655 (print), 1879-2944 (electronic)",
  ISSN-L =       "0010-4655",
  bibdate =      "Mon Dec 2 12:04:56 MST 2013",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/compphyscomm2010.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0010465513002610",
  acknowledgement = ack-nhfb,
  fjournal =     "Computer Physics Communications",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00104655",
}


bibtype: article
filename: elefunt.bib
label: Piparo:2014:SHE
author: Danilo Piparo and Vincenzo Innocente and Thomas Hauth
journal: Journal of Physics: Conference Series
volume: 513
number: 5
publisher: IOP Publishing
pages: 052027
month: jun
monthnumber: 6
year: 2014
DOI: https://doi.org/10.1088/1742-6596/513/5/052027
ISSN: 1742-6588 (print), 1742-6596 (electronic) OR 17426588 (print), 17426596 (electronic)
ISSNL: 1742-6588 OR 17426588
bibdate: Tue Sep 24 14:55:02 2024
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2024.09.24 14:55:02 ???
acknowledgement: ack-nhfb
rawauthor: Danilo Piparo and Vincenzo Innocente and Thomas Hauth
title:
Speeding up HEP experiment software with a library of fast and auto-vectorisable mathematical functions

rawtitle:
Speeding up {HEP} experiment software with a library of fast and auto-vectorisable mathematical functions

entry:
@Article{Piparo:2014:SHE,
  author =       "Danilo Piparo and Vincenzo Innocente and Thomas
                 Hauth",
  title =        "Speeding up {HEP} experiment software with a library
                 of fast and auto-vectorisable mathematical functions",
  journal =      "Journal of Physics: Conference Series",
  volume =       "513",
  number =       "5",
  publisher =    "IOP Publishing",
  pages =        "052027",
  month =        jun,
  year =         "2014",
  DOI =          "https://doi.org/10.1088/1742-6596/513/5/052027",
  ISSN =         "1742-6588 (print), 1742-6596 (electronic)",
  ISSN-L =       "1742-6588",
  bibdate =      "Tue Sep 24 14:55:02 2024",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
}


bibtype: book
filename: elefunt.bib
label: Potter:2014:APC
author: Ronald W. Potter
title: Arbitrary Precision Calculation of Selected Higher Functions
publisher: Lulu
year: 2014
ISBN: 1-312-59943-X OR 131259943X
ISBN13: 978-1-312-59943-7 OR 9781312599437
LCCN: QA76.9.A43 P56 2014
bibdate: Sat Dec 10 15:39:37 MST 2022
bibsource: fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2022.12.10 15:39:37 MST
acknowledgement: ack-nhfb
rawauthor: Ronald W. Potter
rawtitle: Arbitrary Precision Calculation of Selected Higher Functions
subject:
Computer algorithms; Computational complexity; Functional programming languages; Mathematics; Algorithms; Algorithmes; Complexite de calcul (Informatique); Langages de programmation fonctionnels; Mathematiques; algorithms; mathematics; applied mathematics; Algorithms; Computational complexity; Computer algorithms; Functional programming languages; Mathematics

TOC:
Preface / ii Introduction / x 1: Basic Arithmetic / 1-1 2: High Precision Computational Techniques / 2-1 3: Elementary Functions / 3-1 4: Euler's Constant / 4-1 5: Gamma and Polygamma Functions / 5-1 6: Elliptic Integrals and $ pi $ / 6-1 7: Jacobian Elliptic Functions / 7-1 8: Theta Functions / 8-1 9: Incomplete Gamma Functions, Chi$^2$ and Inverse Chi$^2$ Distribution / 9-1 10: Beta and Incomplete Beta Functions, Student's $t$ and $F$-Distributions and Their Inverses / 10-1 11: Error Functions, Gaussian Distribution and Inverse / 11-1 12: Modified Bessel Functions / 12-1 13: Ordinary Bessel Functions / 13-1 14: Zeros of Ordinary Bessel Functions / 14-1 15: Spherical Bessel Functions / 15-1 16: Airy Functions and Zeros / 16-1 17: Kelvin Functions / 17-1 18: Struve Functions / 18-1 19: Fresnel Integrals / 19-1 20: Exponential Integrals / 20-1 21: Sine / Cosine and Sinh / Cosh Integrals / 21-1 22: Orthogonal Polynomials / 22-1 23: Polynomial Roots / 23-1 24: Matrix Operations / 24-1 25: Geometric Operations / 25-1 Appendix A: Fast Fourier Transform (FFT) / A-1 Appendix B: The AGM Algorithm / B-1 Appendix C: Contours of Bessel Function Zeros / C-1 Appendix D: A Few Numbers (6071 digits per number) / D-1 Appendix E: 315061 Decimal Digits of Euler's Constant / E-1 Index / I-1 About the Author

rawTOC:
Preface / ii \\ Introduction / x \\ 1: Basic Arithmetic / 1-1 \\ 2: High Precision Computational Techniques / 2-1 \\ 3: Elementary Functions / 3-1 \\ 4: Euler's Constant / 4-1 \\ 5: Gamma and Polygamma Functions / 5-1 \\ 6: Elliptic Integrals and $ \pi $ / 6-1 \\ 7: Jacobian Elliptic Functions / 7-1 \\ 8: Theta Functions / 8-1 \\ 9: Incomplete Gamma Functions, Chi$^2$ and Inverse Chi$^2$ Distribution / 9-1 \\ 10: Beta and Incomplete Beta Functions, Student's $t$ and $F$-Distributions and Their Inverses / 10-1 \\ 11: Error Functions, Gaussian Distribution and Inverse / 11-1 \\ 12: Modified Bessel Functions / 12-1 \\ 13: Ordinary Bessel Functions / 13-1 \\ 14: Zeros of Ordinary Bessel Functions / 14-1 \\ 15: Spherical Bessel Functions / 15-1 \\ 16: Airy Functions and Zeros / 16-1 \\ 17: Kelvin Functions / 17-1 \\ 18: Struve Functions / 18-1 \\ 19: Fresnel Integrals / 19-1 \\ 20: Exponential Integrals / 20-1 \\ 21: Sine\slash Cosine and Sinh\slash Cosh Integrals / 21-1 \\ 22: Orthogonal Polynomials / 22-1 \\ 23: Polynomial Roots / 23-1 \\ 24: Matrix Operations / 24-1 \\ 25: Geometric Operations / 25-1 \\ Appendix A: Fast Fourier Transform (FFT) / A-1 \\ Appendix B: The AGM Algorithm / B-1 \\ Appendix C: Contours of Bessel Function Zeros / C-1 \\ Appendix D: A Few Numbers (6071 digits per number) / D-1 \\ Appendix E: 315061 Decimal Digits of Euler's Constant / E-1 \\ Index / I-1 \\ About the Author

entry:
@Book{Potter:2014:APC,
  author =       "Ronald W. Potter",
  title =        "Arbitrary Precision Calculation of Selected Higher
                 Functions",
  publisher =    "Lulu",
  address =      "????",
  pages =        "????",
  year =         "2014",
  ISBN =         "1-312-59943-X",
  ISBN-13 =      "978-1-312-59943-7",
  LCCN =         "QA76.9.A43 P56 2014",
  bibdate =      "Sat Dec 10 15:39:37 MST 2022",
  bibsource =    "fsz3950.oclc.org:210/WorldCat;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
  subject =      "Computer algorithms; Computational complexity;
                 Functional programming languages; Mathematics;
                 Algorithms; Algorithmes; Complexit{\'e} de calcul
                 (Informatique); Langages de programmation fonctionnels;
                 Math{\'e}matiques; algorithms; mathematics; applied
                 mathematics; Algorithms; Computational complexity;
                 Computer algorithms; Functional programming languages;
                 Mathematics",
  tableofcontents = "Preface / ii \\
                 Introduction / x \\
                 1: Basic Arithmetic / 1-1 \\
                 2: High Precision Computational Techniques / 2-1 \\
                 3: Elementary Functions / 3-1 \\
                 4: Euler's Constant / 4-1 \\
                 5: Gamma and Polygamma Functions / 5-1 \\
                 6: Elliptic Integrals and $ \pi $ / 6-1 \\
                 7: Jacobian Elliptic Functions / 7-1 \\
                 8: Theta Functions / 8-1 \\
                 9: Incomplete Gamma Functions, Chi$^2$ and Inverse
                 Chi$^2$ Distribution / 9-1 \\
                 10: Beta and Incomplete Beta Functions, Student's $t$
                 and $F$-Distributions and Their Inverses / 10-1 \\
                 11: Error Functions, Gaussian Distribution and Inverse
                 / 11-1 \\
                 12: Modified Bessel Functions / 12-1 \\
                 13: Ordinary Bessel Functions / 13-1 \\
                 14: Zeros of Ordinary Bessel Functions / 14-1 \\
                 15: Spherical Bessel Functions / 15-1 \\
                 16: Airy Functions and Zeros / 16-1 \\
                 17: Kelvin Functions / 17-1 \\
                 18: Struve Functions / 18-1 \\
                 19: Fresnel Integrals / 19-1 \\
                 20: Exponential Integrals / 20-1 \\
                 21: Sine\slash Cosine and Sinh\slash Cosh Integrals /
                 21-1 \\
                 22: Orthogonal Polynomials / 22-1 \\
                 23: Polynomial Roots / 23-1 \\
                 24: Matrix Operations / 24-1 \\
                 25: Geometric Operations / 25-1 \\
                 Appendix A: Fast Fourier Transform (FFT) / A-1 \\
                 Appendix B: The AGM Algorithm / B-1 \\
                 Appendix C: Contours of Bessel Function Zeros / C-1 \\
                 Appendix D: A Few Numbers (6071 digits per number) /
                 D-1 \\
                 Appendix E: 315061 Decimal Digits of Euler's Constant /
                 E-1 \\
                 Index / I-1 \\
                 About the Author",
}


bibtype: article
filename: elefunt.bib
label: Qi:2014:IRC
author: Feng Qi
journal: j-J-COMPUT-APPL-MATH
volume: 268
pages: 155--167
day: 1
month: oct
monthnumber: 10
year: 2014
CODEN: JCAMDI
ISSN: 0377-0427 (print), 1879-1778 (electronic) OR 03770427 (print), 18791778 (electronic)
ISSNL: 0377-0427 OR 03770427
bibdate: Sat Feb 25 13:34:45 MST 2017
bibtimestamp: 2017.02.25 13:34:45 MST
URL: http://www.sciencedirect.com/science/article/pii/S0377042714001356
fjournal: Journal of Computational and Applied Mathematics
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03770427
rawauthor: Feng Qi
title:
Integral representations and complete monotonicity related to the remainder of Burnside's formula for the gamma function

rawtitle:
Integral representations and complete monotonicity related to the remainder of {Burnside}'s formula for the gamma function

entry:
@Article{Qi:2014:IRC,
  author =       "Feng Qi",
  title =        "Integral representations and complete monotonicity
                 related to the remainder of {Burnside}'s formula for
                 the gamma function",
  journal =      j-J-COMPUT-APPL-MATH,
  volume =       "268",
  number =       "??",
  pages =        "155--167",
  day =          "1",
  month =        oct,
  year =         "2014",
  CODEN =        "JCAMDI",
  ISSN =         "0377-0427 (print), 1879-1778 (electronic)",
  ISSN-L =       "0377-0427",
  bibdate =      "Sat Feb 25 13:34:45 MST 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2010.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0377042714001356",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Computational and Applied Mathematics",
  journal-URL =  "http://www.sciencedirect.com/science/journal/03770427",
}


bibtype: inproceedings
filename: elefunt.bib
label: Rappoport:2014:MSM
author: Juri Rappoport
title: Mathematical Software for Modified Bessel Functions
crossref: Hong:2014:MSI
pages: 325--332
year: 2014
DOI: https://doi.org/10.1007/978-3-662-44199-2_51
bibdate: Tue Sep 26 10:17:51 2023
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2023.09.26 10:17:51 ???
acknowledgement: ack-nhfb
rawauthor: Juri Rappoport
rawtitle: Mathematical Software for Modified {Bessel} Functions
entry:
@InProceedings{Rappoport:2014:MSM,
  author =       "Juri Rappoport",
  title =        "Mathematical Software for Modified {Bessel}
                 Functions",
  crossref =     "Hong:2014:MSI",
  pages =        "325--332",
  year =         "2014",
  DOI =          "https://doi.org/10.1007/978-3-662-44199-2_51",
  bibdate =      "Tue Sep 26 10:17:51 2023",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: elefunt.bib
label: Ratnanather:2014:ATI
author: J. Tilak Ratnanather and Jung H. Kim and Sirong Zhang and Anthony M. J. Davis and Stephen K. Lucas
journal: j-TOMS
volume: 40
number: 2
pages: 14:1--14:12
month: feb
monthnumber: 2
year: 2014
CODEN: ACMSCU
DOI: https://doi.org/10.1145/2508435
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
bibdate: Fri Mar 14 06:30:41 MDT 2014
bibtimestamp: 2014.03.14 06:30:41 MDT
fjournal: ACM Transactions on Mathematical Software (TOMS)
acknowledgement: ack-nhfb
articleno: 14
journalURL: https://dl.acm.org/loi/toms
rawauthor: J. Tilak Ratnanather and Jung H. Kim and Sirong Zhang and Anthony M. J. Davis and Stephen K. Lucas
title:
Algorithm 935: tt IIPBF, a tt MATLAB toolbox for infinite integral of products of two Bessel functions

bibsource:
http://www.acm.org/pubs/contents/journals/toms/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/matlab.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib

abstract:
A tt MATLAB toolbox, tt IIPBF, for calculating infinite integrals involving a product of two Bessel functions $ J_a(rho x) J_b(tau x) $, $ J_a(rho x) Y_b(tau x) $, and $ Y_a(rho x) Y_b(tau x) $, for non-negative integers $a$, $b$, and a well-behaved function $ f(x) $, is described. Based on the Lucas algorithm previously developed for $ J_a(rho x) J_b(tau x) $ only, tt IIPBF recasts each product as the sum of two functions whose oscillatory behavior is exploited in the three-step procedure of adaptive integration, summation, and extrapolation. The toolbox uses customised tt QUADPACK and tt IMSL functions from a tt MATLAB conversion of the tt SLATEC library. In addition, tt MATLAB's own tt quadgk function for adaptive Gauss--Kronrod quadrature results in a significant speed up compared with the original algorithm. Usage of tt IIPBF is described and eighteen test cases illustrate the robustness of the toolbox; five additional ones are used to compare tt IIPBF with the tt BESSELINT code for rational and exponential forms of $ f(x) $ with $ J_a(rho x) J_b(tau x) $. Reliability for a broad range of values of $ rho $ and $ tau $ for the three different product types as well as different orders in one case is demonstrated. An electronic appendix provides a novel derivation of formulae for five cases.

rawabstract:
A {\tt MATLAB} toolbox, {\tt IIPBF}, for calculating infinite integrals involving a product of two Bessel functions $ J_a(\rho x) J_b(\tau x) $, $ J_a(\rho x) Y_b(\tau x) $, and $ Y_a(\rho x) Y_b(\tau x) $, for non-negative integers $a$, $b$, and a well-behaved function $ f(x) $, is described. Based on the Lucas algorithm previously developed for $ J_a(\rho x) J_b(\tau x) $ only, {\tt IIPBF} recasts each product as the sum of two functions whose oscillatory behavior is exploited in the three-step procedure of adaptive integration, summation, and extrapolation. The toolbox uses customised {\tt QUADPACK} and {\tt IMSL} functions from a {\tt MATLAB} conversion of the {\tt SLATEC} library. In addition, {\tt MATLAB}'s own {\tt quadgk} function for adaptive Gauss--Kronrod quadrature results in a significant speed up compared with the original algorithm. Usage of {\tt IIPBF} is described and eighteen test cases illustrate the robustness of the toolbox; five additional ones are used to compare {\tt IIPBF} with the {\tt BESSELINT} code for rational and exponential forms of $ f(x) $ with $ J_a(\rho x) J_b(\tau x) $. Reliability for a broad range of values of $ \rho $ and $ \tau $ for the three different product types as well as different orders in one case is demonstrated. An electronic appendix provides a novel derivation of formulae for five cases.

rawtitle:
{Algorithm 935}: {{\tt IIPBF}}, a {{\tt MATLAB}} toolbox for infinite integral of products of two {Bessel} functions

entry:
@Article{Ratnanather:2014:ATI,
  author =       "J. Tilak Ratnanather and Jung H. Kim and Sirong Zhang
                 and Anthony M. J. Davis and Stephen K. Lucas",
  title =        "{Algorithm 935}: {{\tt IIPBF}}, a {{\tt MATLAB}}
                 toolbox for infinite integral of products of two
                 {Bessel} functions",
  journal =      j-TOMS,
  volume =       "40",
  number =       "2",
  pages =        "14:1--14:12",
  month =        feb,
  year =         "2014",
  CODEN =        "ACMSCU",
  DOI =          "https://doi.org/10.1145/2508435",
  ISSN =         "0098-3500 (print), 1557-7295 (electronic)",
  ISSN-L =       "0098-3500",
  bibdate =      "Fri Mar 14 06:30:41 MDT 2014",
  bibsource =    "http://www.acm.org/pubs/contents/journals/toms/;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/matlab.bib;
                 https://www.math.utah.edu/pub/tex/bib/toms.bib",
  abstract =     "A {\tt MATLAB} toolbox, {\tt IIPBF}, for calculating
                 infinite integrals involving a product of two Bessel
                 functions $ J_a(\rho x) J_b(\tau x) $, $ J_a(\rho x)
                 Y_b(\tau x) $, and $ Y_a(\rho x) Y_b(\tau x) $, for
                 non-negative integers $a$, $b$, and a well-behaved
                 function $ f(x) $, is described. Based on the Lucas
                 algorithm previously developed for $ J_a(\rho x)
                 J_b(\tau x) $ only, {\tt IIPBF} recasts each product as
                 the sum of two functions whose oscillatory behavior is
                 exploited in the three-step procedure of adaptive
                 integration, summation, and extrapolation. The toolbox
                 uses customised {\tt QUADPACK} and {\tt IMSL} functions
                 from a {\tt MATLAB} conversion of the {\tt SLATEC}
                 library. In addition, {\tt MATLAB}'s own {\tt quadgk}
                 function for adaptive Gauss--Kronrod quadrature results
                 in a significant speed up compared with the original
                 algorithm. Usage of {\tt IIPBF} is described and
                 eighteen test cases illustrate the robustness of the
                 toolbox; five additional ones are used to compare {\tt
                 IIPBF} with the {\tt BESSELINT} code for rational and
                 exponential forms of $ f(x) $ with $ J_a(\rho x)
                 J_b(\tau x) $. Reliability for a broad range of values
                 of $ \rho $ and $ \tau $ for the three different
                 product types as well as different orders in one case
                 is demonstrated. An electronic appendix provides a
                 novel derivation of formulae for five cases.",
  acknowledgement = ack-nhfb,
  articleno =    "14",
  fjournal =     "ACM Transactions on Mathematical Software (TOMS)",
  journal-URL =  "https://dl.acm.org/loi/toms",
}


bibtype: article
filename: elefunt.bib
label: Shukla:2014:LLH
author: R. Shukla and K. C. Ray
title: Low Latency Hybrid CORDIC Algorithm
journal: j-IEEE-TRANS-COMPUT
volume: 63
number: 12
pages: 3066--3078
month: dec
monthnumber: 12
year: 2014
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2013.173
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Thu Dec 4 10:36:57 2014
bibtimestamp: 2014.12.04 10:36:57 ???
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: R. Shukla and K. C. Ray
rawtitle: Low Latency Hybrid {CORDIC} Algorithm
bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib

keywords:
adders; Approximation algorithms; communication systems; Computer architecture; coordinate rotational digital computer; CORDIC algorithm; Delays; digital arithmetic; Digital computers; digital computers; double step branching; fast adders; first order hardware architecture; hardware complexity; hybrid CORDIC algorithm; image processing; low latency; low latency hybrid CORDIC algorithm; Mathematical model; radix-4; redundant arithmetic; scale factor calculation; signal processing; Signal processing algorithms

entry:
@Article{Shukla:2014:LLH,
  author =       "R. Shukla and K. C. Ray",
  title =        "Low Latency Hybrid {CORDIC} Algorithm",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "63",
  number =       "12",
  pages =        "3066--3078",
  month =        dec,
  year =         "2014",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2013.173",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Thu Dec 4 10:36:57 2014",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
  keywords =     "adders; Approximation algorithms; communication
                 systems; Computer architecture; coordinate rotational
                 digital computer; CORDIC algorithm; Delays; digital
                 arithmetic; Digital computers; digital computers;
                 double step branching; fast adders; first order
                 hardware architecture; hardware complexity; hybrid
                 CORDIC algorithm; image processing; low latency; low
                 latency hybrid CORDIC algorithm; Mathematical model;
                 radix-4; redundant arithmetic; scale factor
                 calculation; signal processing; Signal processing
                 algorithms",
}


bibtype: article
filename: elefunt.bib
label: Soranzo:2014:VSE
author: Alessandro Soranzo and Emanuela Epure
title: Very simply explicitly invertible approximations of normal cumulative and normal quantile function
journal: j-APPL-MATH-SCI-RUSE
volume: 8
pages: 4323--4341
year: 2014
DOI: https://doi.org/10.12988/ams.2014.45338
ISSN: 1312-885X (print), 1314-7552 (electronic) OR 1312885X (print), 13147552 (electronic)
ISSNL: 1312-885X OR 1312885X
bibdate: Sat Dec 16 17:41:14 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2017.12.16 17:41:14 ???
URL: http://m-hikari.com/ams/ams-2014/ams-85-88-2014/epureAMS85-88-2014.pdf
fjournal: Applied Mathematical Sciences (Ruse)
acknowledgement: ack-nhfb
journalURL: http://www.m-hikari.com/ams/
rawauthor: Alessandro Soranzo and Emanuela Epure
rawtitle: Very simply explicitly invertible approximations of normal cumulative and normal quantile function
entry:
@Article{Soranzo:2014:VSE,
  author =       "Alessandro Soranzo and Emanuela Epure",
  title =        "Very simply explicitly invertible approximations of
                 normal cumulative and normal quantile function",
  journal =      j-APPL-MATH-SCI-RUSE,
  volume =       "8",
  pages =        "4323--4341",
  year =         "2014",
  DOI =          "https://doi.org/10.12988/ams.2014.45338",
  ISSN =         "1312-885X (print), 1314-7552 (electronic)",
  ISSN-L =       "1312-885X",
  bibdate =      "Sat Dec 16 17:41:14 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://m-hikari.com/ams/ams-2014/ams-85-88-2014/epureAMS85-88-2014.pdf",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied Mathematical Sciences (Ruse)",
  journal-URL =  "http://www.m-hikari.com/ams/",
}


bibtype: article
filename: elefunt.bib
label: Wang:2014:CFA
author: Dong Wang and Milos D. Ercegovac and Yang Xiao
title: Complex Function Approximation Using Two-Dimensional Interpolation
journal: j-IEEE-TRANS-COMPUT
volume: 63
number: 12
pages: 2948--2960
month: dec
monthnumber: 12
year: 2014
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2013.181
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Thu Dec 4 10:36:57 2014
bibtimestamp: 2014.12.04 10:36:57 ???
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: Dong Wang and Milo{\v{s}} D. Ercegovac and Yang Xiao
rawtitle: Complex Function Approximation Using Two-Dimensional Interpolation
keywords:
2D convolution algorithm; 2D interpolation; Approximation error; ASIC; bipartite schemes; bivariate functions; coefficient table; complex exponential; complex function approximation; complex function evaluation; complex reciprocal; Complex reciprocal; Computational complexity; cubic interpolation; exponential functions; field programmable gate arrays; FPGA; Function approximation; generic hardware architecture; interpolation; interpolation degree; interpolation kernels; Lagrange interpolation; Lagrangian functions; linear interpolation; lookup tables; memory requirements; multipartite schemes; quadratic interpolation; Quadratic programming; table lookup; tabulated function; two-dimensional interpolation

entry:
@Article{Wang:2014:CFA,
  author =       "Dong Wang and Milo{\v{s}} D. Ercegovac and Yang Xiao",
  title =        "Complex Function Approximation Using Two-Dimensional
                 Interpolation",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "63",
  number =       "12",
  pages =        "2948--2960",
  month =        dec,
  year =         "2014",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2013.181",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Thu Dec 4 10:36:57 2014",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput2010.bib",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
  keywords =     "2D convolution algorithm; 2D interpolation;
                 Approximation error; ASIC; bipartite schemes; bivariate
                 functions; coefficient table; complex exponential;
                 complex function approximation; complex function
                 evaluation; complex reciprocal; Complex reciprocal;
                 Computational complexity; cubic interpolation;
                 exponential functions; field programmable gate arrays;
                 FPGA; Function approximation; generic hardware
                 architecture; interpolation; interpolation degree;
                 interpolation kernels; Lagrange interpolation;
                 Lagrangian functions; linear interpolation; lookup
                 tables; memory requirements; multipartite schemes;
                 quadratic interpolation; Quadratic programming; table
                 lookup; tabulated function; two-dimensional
                 interpolation",
}


bibtype: article
filename: elefunt.bib
label: Wang:2014:FPT
author: Dong Wang and Jean-Michel Muller and Nicolas Brisebarre and Milo D. Ercegovac
title: $ (M, p, k)$-Friendly Points: a Table-Based Method to Evaluate Trigonometric Function
journal: j-IEEE-TRANS-CIRCUITS-SYST-II-EXPRESS-BRIEFS
volume: 61
number: 9
pages: 711--715
year: 2014
DOI: https://doi.org/10.1109/TCSII.2014.2331094
ISSN: 1549-7747 (print), 1558-3791 (electronic) OR 15497747 (print), 15583791 (electronic)
ISSNL: 1549-7747 OR 15497747
bibdate: Fri Sep 29 10:46:18 2023
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2023.09.29 10:46:18 ???
fjournal: IEEE Transactions on Circuits and Systems II: Express Briefs
acknowledgement: ack-nhfb
journalURL: https://ieeexplore.ieee.org/xpl/issues?punumber=8920
rawauthor: Dong Wang and Jean-Michel Muller and Nicolas Brisebarre and Milo D. Ercegovac
rawtitle: {$ (M, p, k)$-Friendly} Points: a Table-Based Method to Evaluate Trigonometric Function
entry:
@Article{Wang:2014:FPT,
  author =       "Dong Wang and Jean-Michel Muller and Nicolas
                 Brisebarre and Milo D. Ercegovac",
  title =        "{$ (M, p, k)$-Friendly} Points: a Table-Based Method
                 to Evaluate Trigonometric Function",
  journal =      j-IEEE-TRANS-CIRCUITS-SYST-II-EXPRESS-BRIEFS,
  volume =       "61",
  number =       "9",
  pages =        "711--715",
  year =         "2014",
  DOI =          "https://doi.org/10.1109/TCSII.2014.2331094",
  ISSN =         "1549-7747 (print), 1558-3791 (electronic)",
  ISSN-L =       "1549-7747",
  bibdate =      "Fri Sep 29 10:46:18 2023",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "IEEE Transactions on Circuits and Systems II: Express
                 Briefs",
  journal-URL =  "https://ieeexplore.ieee.org/xpl/issues?punumber=8920",
}


bibtype: article
filename: elefunt.bib
label: Xu:2014:SII
author: Ai-Min Xu and Zhong-Di Cen
title: Some identities involving exponential functions and Stirling numbers and applications
journal: j-J-COMPUT-APPL-MATH
volume: 260
pages: 201--207
month: apr
monthnumber: 4
year: 2014
CODEN: JCAMDI
ISSN: 0377-0427 (print), 1879-1778 (electronic) OR 03770427 (print), 18791778 (electronic)
ISSNL: 0377-0427 OR 03770427
bibdate: Sat Feb 25 13:34:42 MST 2017
bibtimestamp: 2017.02.25 13:34:42 MST
URL: http://www.sciencedirect.com/science/article/pii/S0377042713005323
fjournal: Journal of Computational and Applied Mathematics
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03770427
rawauthor: Ai-Min Xu and Zhong-Di Cen
rawtitle: Some identities involving exponential functions and {Stirling} numbers and applications
entry:
@Article{Xu:2014:SII,
  author =       "Ai-Min Xu and Zhong-Di Cen",
  title =        "Some identities involving exponential functions and
                 {Stirling} numbers and applications",
  journal =      j-J-COMPUT-APPL-MATH,
  volume =       "260",
  number =       "??",
  pages =        "201--207",
  month =        apr,
  year =         "2014",
  CODEN =        "JCAMDI",
  ISSN =         "0377-0427 (print), 1879-1778 (electronic)",
  ISSN-L =       "0377-0427",
  bibdate =      "Sat Feb 25 13:34:42 MST 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2010.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0377042713005323",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Computational and Applied Mathematics",
  journal-URL =  "http://www.sciencedirect.com/science/journal/03770427",
}


bibtype: article
filename: elefunt.bib
label: Yun:2014:AHA
author: Beong In Yun
title: An ad hoc approximation to the Gauss error function and a correction method
journal: j-APPL-MATH-SCI-RUSE
volume: 8
pages: 4261--4273
year: 2014
DOI: https://doi.org/10.12988/ams.2014.45345
ISSN: 1312-885X (print), 1314-7552 (electronic) OR 1312885X (print), 13147552 (electronic)
bibdate: Sat Dec 16 18:06:18 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2017.12.16 18:06:18 ???
URL: http://www.m-hikari.com/ams/ams-2014/ams-85-88-2014/yunAMS85-88-2014.pdf
fjournal: Applied Mathematical Sciences (Ruse)
acknowledgement: ack-nhfb
journalURL: http://www.m-hikari.com/ams/
rawauthor: Beong In Yun
rawtitle: An ad hoc approximation to the {Gauss} error function and a correction method
entry:
@Article{Yun:2014:AHA,
  author =       "Beong In Yun",
  title =        "An ad hoc approximation to the {Gauss} error function
                 and a correction method",
  journal =      j-APPL-MATH-SCI-RUSE,
  volume =       "8",
  pages =        "4261--4273",
  year =         "2014",
  DOI =          "https://doi.org/10.12988/ams.2014.45345",
  ISSN =         "1312-885X (print), 1314-7552 (electronic)",
  bibdate =      "Sat Dec 16 18:06:18 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.m-hikari.com/ams/ams-2014/ams-85-88-2014/yunAMS85-88-2014.pdf",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied Mathematical Sciences (Ruse)",
  journal-URL =  "http://www.m-hikari.com/ams/",
}


bibtype: inproceedings
filename: elefunt.bib
label: Zafar:2014:HAD
author: Saad Zafar and Raviteja Adapa
booktitle: 2014 International Conference on Advances in Electrical Engineering (ICAEE)
title: Hardware architecture design and mapping of ``Fast Inverse Square Root'' algorithm
publisher: pub-IEEE
address: pub-IEEE:adr
pages: 1--4
month: jan
monthnumber: 1
year: 2014
DOI: https://doi.org/10.1109/icaee.2014.6838433
bibdate: Wed Dec 20 07:29:37 2023
bibtimestamp: 2023.12.20 07:29:37 ???
acknowledgement: ack-nhfb
rawauthor: Saad Zafar and Raviteja Adapa
rawbooktitle: 2014 International Conference on Advances in Electrical Engineering {(ICAEE)}
rawtitle: Hardware architecture design and mapping of ``{Fast Inverse Square Root}'' algorithm
abstract:
The Fast Inverse Square Root algorithm has been used in 3D games of past for lighting and reflection calculations, because it offers up to four times performance gains. This paper presents a hardware implementation of the algorithm on an FPGA board by designing the complete architecture and successfully mapping it on Xilinx Spartan 3E after thorough functional verification. The results show that this implementation provides a very efficient single-precision floating point inverse square root calculator with practically accurate results being made available after just 12 short clock cycles. This performance measure is far superior to the software counterpart of the algorithm, and is not processor dependent like rsqrtss of x86 SSE instruction set. Results of this work can aid FPGA based vector processors or graphic processing units with 3D rendering. The hardware design can also form part of a larger floating point arithmetic unit for dedicated reciprocal square root calculations.

rawabstract:
The Fast Inverse Square Root algorithm has been used in 3D games of past for lighting and reflection calculations, because it offers up to four times performance gains. This paper presents a hardware implementation of the algorithm on an FPGA board by designing the complete architecture and successfully mapping it on Xilinx Spartan 3E after thorough functional verification. The results show that this implementation provides a very efficient single-precision floating point inverse square root calculator with practically accurate results being made available after just 12 short clock cycles. This performance measure is far superior to the software counterpart of the algorithm, and is not processor dependent like rsqrtss of x86 SSE instruction set. Results of this work can aid FPGA based vector processors or graphic processing units with 3D rendering. The hardware design can also form part of a larger floating point arithmetic unit for dedicated reciprocal square root calculations.

entry:
@InProceedings{Zafar:2014:HAD,
  author =       "Saad Zafar and Raviteja Adapa",
  booktitle =    "2014 International Conference on Advances in
                 Electrical Engineering {(ICAEE)}",
  title =        "Hardware architecture design and mapping of ``{Fast
                 Inverse Square Root}'' algorithm",
  publisher =    pub-IEEE,
  address =      pub-IEEE:adr,
  pages =        "1--4",
  month =        jan,
  year =         "2014",
  DOI =          "https://doi.org/10.1109/icaee.2014.6838433",
  bibdate =      "Wed Dec 20 07:29:37 2023",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  abstract =     "The Fast Inverse Square Root algorithm has been used
                 in 3D games of past for lighting and reflection
                 calculations, because it offers up to four times
                 performance gains. This paper presents a hardware
                 implementation of the algorithm on an FPGA board by
                 designing the complete architecture and successfully
                 mapping it on Xilinx Spartan 3E after thorough
                 functional verification. The results show that this
                 implementation provides a very efficient
                 single-precision floating point inverse square root
                 calculator with practically accurate results being made
                 available after just 12 short clock cycles. This
                 performance measure is far superior to the software
                 counterpart of the algorithm, and is not processor
                 dependent like rsqrtss of x86 SSE instruction set.
                 Results of this work can aid FPGA based vector
                 processors or graphic processing units with 3D
                 rendering. The hardware design can also form part of a
                 larger floating point arithmetic unit for dedicated
                 reciprocal square root calculations.",
  acknowledgement = ack-nhfb,
}


bibtype: misc
filename: elefunt.bib
label: Anonymous:2015:L
author: Anonymous
title: libcerf
pages: 1--4
year: 2015
bibdate: Mon Jun 12 16:08:24 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2017.06.12 16:08:24 ???
URL: http://apps.jcns.fz-juelich.de/doku/sc/libcerf
acknowledgement: ack-nhfb
howpublished: Web site
rawauthor: Anonymous
rawtitle: libcerf
abstract:
This is the home page of libcerf, a self-contained numeric library that provides an efficient and accurate implementation of complex error functions, along with Dawson, Faddeeva, and Voigt functions.

rawabstract:
This is the home page of libcerf, a self-contained numeric library that provides an efficient and accurate implementation of complex error functions, along with Dawson, Faddeeva, and Voigt functions.

entry:
@Misc{Anonymous:2015:L,
  author =       "Anonymous",
  title =        "libcerf",
  howpublished = "Web site",
  year =         "2015",
  bibdate =      "Mon Jun 12 16:08:24 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://apps.jcns.fz-juelich.de/doku/sc/libcerf",
  abstract =     "This is the home page of libcerf, a self-contained
                 numeric library that provides an efficient and accurate
                 implementation of complex error functions, along with
                 Dawson, Faddeeva, and Voigt functions.",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: elefunt.bib
label: Bailey:2015:CCI
author: D. H. Bailey and J. M. Borwein
journal: j-APPL-MATH-COMP
volume: 268
pages: 462--477
day: 1
month: oct
monthnumber: 10
year: 2015
CODEN: AMHCBQ
ISSN: 0096-3003 (print), 1873-5649 (electronic) OR 00963003 (print), 18735649 (electronic)
ISSNL: 0096-3003 OR 00963003
bibdate: Wed Sep 16 06:56:32 MDT 2015
bibtimestamp: 2015.09.16 06:56:32 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0096300315008292
fjournal: Applied Mathematics and Computation
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00963003/
rawauthor: D. H. Bailey and J. M. Borwein
title:
Crandall's computation of the incomplete Gamma function and the Hurwitz zeta function, with applications to Dirichlet $L$-series

bibsource:
https://www.math.utah.edu/pub/bibnet/authors/c/crandall-richard-e.bib; https://www.math.utah.edu/pub/tex/bib/applmathcomput2015.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

rawtitle:
{Crandall}'s computation of the incomplete Gamma function and the {Hurwitz} zeta function, with applications to {Dirichlet} {$L$}-series

entry:
@Article{Bailey:2015:CCI,
  author =       "D. H. Bailey and J. M. Borwein",
  title =        "{Crandall}'s computation of the incomplete Gamma
                 function and the {Hurwitz} zeta function, with
                 applications to {Dirichlet} {$L$}-series",
  journal =      j-APPL-MATH-COMP,
  volume =       "268",
  number =       "??",
  pages =        "462--477",
  day =          "1",
  month =        oct,
  year =         "2015",
  CODEN =        "AMHCBQ",
  ISSN =         "0096-3003 (print), 1873-5649 (electronic)",
  ISSN-L =       "0096-3003",
  bibdate =      "Wed Sep 16 06:56:32 MDT 2015",
  bibsource =    "https://www.math.utah.edu/pub/bibnet/authors/c/crandall-richard-e.bib;
                 https://www.math.utah.edu/pub/tex/bib/applmathcomput2015.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0096300315008292",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied Mathematics and Computation",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00963003/",
}


bibtype: article
filename: elefunt.bib
label: Beliakov:2015:ARZ
author: Gleb Beliakov and Yuri Matiyasevich
journal: j-EXP-MATH
volume: 24
number: 2
pages: 150--161
year: 2015
DOI: https://doi.org/10.1080/10586458.2014.976801
ISSN: 1058-6458 (print), 1944-950X (electronic) OR 10586458 (print), 1944950X (electronic)
ISSNL: 1058-6458 OR 10586458
bibdate: Mon Jun 8 17:49:44 MDT 2015
bibtimestamp: 2015.06.08 17:49:44 MDT
fjournal: Experimental Mathematics
acknowledgement: ack-nhfb
journalURL: http://www.tandfonline.com/loi/uexm20
rawauthor: Gleb Beliakov and Yuri Matiyasevich
title:
Approximation of Riemann's Zeta Function by Finite Dirichlet Series: A Multiprecision Numerical Approach

bibsource:
http://www.tandfonline.com/toc/uexm20/24/2; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/expmath.bib

rawtitle:
Approximation of {Riemann}'s Zeta Function by Finite {Dirichlet} Series: A Multiprecision Numerical Approach

entry:
@Article{Beliakov:2015:ARZ,
  author =       "Gleb Beliakov and Yuri Matiyasevich",
  title =        "Approximation of {Riemann}'s Zeta Function by Finite
                 {Dirichlet} Series: A Multiprecision Numerical
                 Approach",
  journal =      j-EXP-MATH,
  volume =       "24",
  number =       "2",
  pages =        "150--161",
  year =         "2015",
  CODEN =        "????",
  DOI =          "https://doi.org/10.1080/10586458.2014.976801",
  ISSN =         "1058-6458 (print), 1944-950X (electronic)",
  ISSN-L =       "1058-6458",
  bibdate =      "Mon Jun 8 17:49:44 MDT 2015",
  bibsource =    "http://www.tandfonline.com/toc/uexm20/24/2;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/expmath.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "Experimental Mathematics",
  journal-URL =  "http://www.tandfonline.com/loi/uexm20",
}


bibtype: article
filename: elefunt.bib
label: Boyd:2015:FWC
author: John P. Boyd
title: Four ways to compute the inverse of the complete elliptic integral of the first kind
journal: j-COMP-PHYS-COMM
volume: 196
pages: 13--18
month: nov
monthnumber: 11
year: 2015
CODEN: CPHCBZ
DOI: https://doi.org/10.1016/j.cpc.2015.05.006
ISSN: 0010-4655 (print), 1879-2944 (electronic) OR 00104655 (print), 18792944 (electronic)
ISSNL: 0010-4655 OR 00104655
bibdate: Tue Sep 22 13:45:19 MDT 2015
bibtimestamp: 2015.09.22 13:45:19 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0010465515001733
fjournal: Computer Physics Communications
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00104655/
rawauthor: John P. Boyd
rawtitle: Four ways to compute the inverse of the complete elliptic integral of the first kind
bibsource:
https://www.math.utah.edu/pub/tex/bib/compphyscomm2010.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/maple-extract.bib; https://www.math.utah.edu/pub/tex/bib/matlab.bib

entry:
@Article{Boyd:2015:FWC,
  author =       "John P. Boyd",
  title =        "Four ways to compute the inverse of the complete
                 elliptic integral of the first kind",
  journal =      j-COMP-PHYS-COMM,
  volume =       "196",
  number =       "??",
  pages =        "13--18",
  month =        nov,
  year =         "2015",
  CODEN =        "CPHCBZ",
  DOI =          "https://doi.org/10.1016/j.cpc.2015.05.006",
  ISSN =         "0010-4655 (print), 1879-2944 (electronic)",
  ISSN-L =       "0010-4655",
  bibdate =      "Tue Sep 22 13:45:19 MDT 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/compphyscomm2010.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/maple-extract.bib;
                 https://www.math.utah.edu/pub/tex/bib/matlab.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0010465515001733",
  acknowledgement = ack-nhfb,
  fjournal =     "Computer Physics Communications",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00104655/",
}


bibtype: article
filename: elefunt.bib
label: Brent:2015:BET
author: Richard P. Brent and Fredrik Johansson
title: A bound for the error term in the Brent--McMillan algorithm
journal: j-MATH-COMPUT
volume: 84
number: 295
pages: 2351--2359
year: 2015
CODEN: MCMPAF
ISSN: 0025-5718 (print), 1088-6842 (electronic) OR 00255718 (print), 10886842 (electronic)
ISSNL: 0025-5718 OR 00255718
bibdate: Tue Aug 4 08:33:55 MDT 2015
bibtimestamp: 2015.08.04 08:33:55 MDT
fjournal: Mathematics of Computation
keywords: Euler's constant; multiple-precision arithmetic
acknowledgement: ack-nhfb
journalURL: http://www.ams.org/mcom/
rawauthor: Richard P. Brent and Fredrik Johansson
rawtitle: A bound for the error term in the {Brent--McMillan} algorithm
bibsource:
http://www.ams.org/mcom/2015-84-295; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp2010.bib

URL:
http://www.ams.org/journals/mcom/2015-84-295/S0025-5718-2015-02931-7; http://www.ams.org/journals/mcom/2015-84-295/S0025-5718-2015-02931-7/S0025-5718-2015-02931-7.pdf

abstract:
The Brent--McMillan algorithm B3 (1980), when implemented with binary splitting, is the fastest known algorithm for high-precision computation of Euler's constant. However, no rigorous error bound for the algorithm has ever been published. We provide such a bound and justify the empirical observations of Brent and McMillan. We also give bounds on the error in the asymptotic expansions of functions related to the Bessel functions $ I_0 (x) $ and $ K_0 (x) $ for positive real $x$.

rawabstract:
The Brent--McMillan algorithm B3 (1980), when implemented with binary splitting, is the fastest known algorithm for high-precision computation of Euler's constant. However, no rigorous error bound for the algorithm has ever been published. We provide such a bound and justify the empirical observations of Brent and McMillan. We also give bounds on the error in the asymptotic expansions of functions related to the Bessel functions $ I_0 (x) $ and $ K_0 (x) $ for positive real $x$.

entry:
@Article{Brent:2015:BET,
  author =       "Richard P. Brent and Fredrik Johansson",
  title =        "A bound for the error term in the {Brent--McMillan}
                 algorithm",
  journal =      j-MATH-COMPUT,
  volume =       "84",
  number =       "295",
  pages =        "2351--2359",
  month =        "",
  year =         "2015",
  CODEN =        "MCMPAF",
  ISSN =         "0025-5718 (print), 1088-6842 (electronic)",
  ISSN-L =       "0025-5718",
  bibdate =      "Tue Aug 4 08:33:55 MDT 2015",
  bibsource =    "http://www.ams.org/mcom/2015-84-295;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/mathcomp2010.bib",
  URL =          "http://www.ams.org/journals/mcom/2015-84-295/S0025-5718-2015-02931-7;
                 http://www.ams.org/journals/mcom/2015-84-295/S0025-5718-2015-02931-7/S0025-5718-2015-02931-7.pdf",
  abstract =     "The Brent--McMillan algorithm B3 (1980), when
                 implemented with binary splitting, is the fastest known
                 algorithm for high-precision computation of Euler's
                 constant. However, no rigorous error bound for the
                 algorithm has ever been published. We provide such a
                 bound and justify the empirical observations of Brent
                 and McMillan. We also give bounds on the error in the
                 asymptotic expansions of functions related to the
                 Bessel functions $ I_0 (x) $ and $ K_0 (x) $ for
                 positive real $x$.",
  acknowledgement = ack-nhfb,
  fjournal =     "Mathematics of Computation",
  journal-URL =  "http://www.ams.org/mcom/",
  keywords =     "Euler's constant; multiple-precision arithmetic",
}


bibtype: inproceedings
filename: elefunt.bib
label: Brunie:2015:CGM
author: Nicolas Brunie and Florent de Dinechin and Olga Kupriianova and Christoph Lauter
title: Code Generators for Mathematical Functions
crossref: Muller:2015:ISC
pages: 66--73
year: 2015
DOI: https://doi.org/10.1109/ARITH.2015.22
bibdate: Sat Aug 01 08:05:52 2015
bibtimestamp: 2015.08.01 08:05:52 ???
keywords: ARITH-22
acknowledgement: ack-nhfb
rawauthor: Nicolas Brunie and Florent de Dinechin and Olga Kupriianova and Christoph Lauter
rawtitle: Code Generators for Mathematical Functions
entry:
@InProceedings{Brunie:2015:CGM,
  author =       "Nicolas Brunie and Florent de Dinechin and Olga
                 Kupriianova and Christoph Lauter",
  title =        "Code Generators for Mathematical Functions",
  crossref =     "Muller:2015:ISC",
  pages =        "66--73",
  year =         "2015",
  DOI =          "https://doi.org/10.1109/ARITH.2015.22",
  bibdate =      "Sat Aug 01 08:05:52 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH-22",
}


bibtype: article
filename: elefunt.bib
label: Chen:2015:AEC
author: Chao-Ping Chen and Neven Elezovic
journal: j-APPL-MATH-COMP
volume: 269
pages: 232--241
day: 15
month: oct
monthnumber: 10
year: 2015
CODEN: AMHCBQ
ISSN: 0096-3003 (print), 1873-5649 (electronic) OR 00963003 (print), 18735649 (electronic)
ISSNL: 0096-3003 OR 00963003
bibdate: Wed Sep 16 06:56:33 MDT 2015
bibtimestamp: 2015.09.16 06:56:33 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0096300315009637
fjournal: Applied Mathematics and Computation
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00963003/
rawauthor: Chao-Ping Chen and Neven Elezovi{\'c}
title:
Asymptotic expansions and completely monotonic functions associated with the gamma, psi and polygamma functions

bibsource:
https://www.math.utah.edu/pub/tex/bib/applmathcomput2015.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

rawtitle:
Asymptotic expansions and completely monotonic functions associated with the gamma, psi and polygamma functions

entry:
@Article{Chen:2015:AEC,
  author =       "Chao-Ping Chen and Neven Elezovi{\'c}",
  title =        "Asymptotic expansions and completely monotonic
                 functions associated with the gamma, psi and polygamma
                 functions",
  journal =      j-APPL-MATH-COMP,
  volume =       "269",
  number =       "??",
  pages =        "232--241",
  day =          "15",
  month =        oct,
  year =         "2015",
  CODEN =        "AMHCBQ",
  ISSN =         "0096-3003 (print), 1873-5649 (electronic)",
  ISSN-L =       "0096-3003",
  bibdate =      "Wed Sep 16 06:56:33 MDT 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/applmathcomput2015.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0096300315009637",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied Mathematics and Computation",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00963003/",
}


bibtype: article
filename: elefunt.bib
label: Chen:2015:IAEa
author: Chao-Ping Chen and Richard B. Paris
journal: j-APPL-MATH-COMP
volume: 250
pages: 514--529
day: 1
month: jan
monthnumber: 1
year: 2015
CODEN: AMHCBQ
ISSN: 0096-3003 (print), 1873-5649 (electronic) OR 00963003 (print), 18735649 (electronic)
ISSNL: 0096-3003 OR 00963003
bibdate: Wed Jan 7 16:27:08 MST 2015
bibtimestamp: 2015.01.07 16:27:08 MST
URL: http://www.sciencedirect.com/science/article/pii/S009630031401515X
fjournal: Applied Mathematics and Computation
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00963003/
rawauthor: Chao-Ping Chen and Richard B. Paris
title:
Inequalities, asymptotic expansions and completely monotonic functions related to the gamma function

bibsource:
https://www.math.utah.edu/pub/tex/bib/applmathcomput2015.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

rawtitle:
Inequalities, asymptotic expansions and completely monotonic functions related to the gamma function

entry:
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  author =       "Chao-Ping Chen and Richard B. Paris",
  title =        "Inequalities, asymptotic expansions and completely
                 monotonic functions related to the gamma function",
  journal =      j-APPL-MATH-COMP,
  volume =       "250",
  number =       "??",
  pages =        "514--529",
  day =          "1",
  month =        jan,
  year =         "2015",
  CODEN =        "AMHCBQ",
  ISSN =         "0096-3003 (print), 1873-5649 (electronic)",
  ISSN-L =       "0096-3003",
  bibdate =      "Wed Jan 7 16:27:08 MST 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/applmathcomput2015.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S009630031401515X",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied Mathematics and Computation",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00963003/",
}


bibtype: article
filename: elefunt.bib
label: Chen:2015:IAEb
author: Chao-Ping Chen
journal: j-APPL-MATH-COMP
volume: 261
pages: 337--350
day: 15
month: jun
monthnumber: 6
year: 2015
CODEN: AMHCBQ
ISSN: 0096-3003 (print), 1873-5649 (electronic) OR 00963003 (print), 18735649 (electronic)
ISSNL: 0096-3003 OR 00963003
bibdate: Wed May 13 09:01:41 MDT 2015
bibtimestamp: 2015.05.13 09:01:41 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0096300315004610
fjournal: Applied Mathematics and Computation
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00963003/
rawauthor: Chao-Ping Chen
title:
Inequalities and asymptotic expansions associated with the Ramanujan and Nemes formulas for the gamma function

bibsource:
https://www.math.utah.edu/pub/tex/bib/applmathcomput2015.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

rawtitle:
Inequalities and asymptotic expansions associated with the {Ramanujan} and {Nemes} formulas for the gamma function

entry:
@Article{Chen:2015:IAEb,
  author =       "Chao-Ping Chen",
  title =        "Inequalities and asymptotic expansions associated with
                 the {Ramanujan} and {Nemes} formulas for the gamma
                 function",
  journal =      j-APPL-MATH-COMP,
  volume =       "261",
  number =       "??",
  pages =        "337--350",
  day =          "15",
  month =        jun,
  year =         "2015",
  CODEN =        "AMHCBQ",
  ISSN =         "0096-3003 (print), 1873-5649 (electronic)",
  ISSN-L =       "0096-3003",
  bibdate =      "Wed May 13 09:01:41 MDT 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/applmathcomput2015.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0096300315004610",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied Mathematics and Computation",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00963003/",
}


bibtype: article
filename: elefunt.bib
label: Chen:2015:ICM
author: Chao-Ping Chen
journal: j-APPL-MATH-COMP
volume: 259
pages: 790--799
day: 15
month: may
monthnumber: 5
year: 2015
CODEN: AMHCBQ
ISSN: 0096-3003 (print), 1873-5649 (electronic) OR 00963003 (print), 18735649 (electronic)
ISSNL: 0096-3003 OR 00963003
bibdate: Fri Apr 24 18:27:24 MDT 2015
bibtimestamp: 2015.04.24 18:27:24 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0096300315003148
fjournal: Applied Mathematics and Computation
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00963003/
rawauthor: Chao-Ping Chen
title:
Inequalities and completely monotonic functions associated with the ratios of functions resulting from the gamma function

rawtitle:
Inequalities and completely monotonic functions associated with the ratios of functions resulting from the gamma function

entry:
@Article{Chen:2015:ICM,
  author =       "Chao-Ping Chen",
  title =        "Inequalities and completely monotonic functions
                 associated with the ratios of functions resulting from
                 the gamma function",
  journal =      j-APPL-MATH-COMP,
  volume =       "259",
  number =       "??",
  pages =        "790--799",
  day =          "15",
  month =        may,
  year =         "2015",
  CODEN =        "AMHCBQ",
  ISSN =         "0096-3003 (print), 1873-5649 (electronic)",
  ISSN-L =       "0096-3003",
  bibdate =      "Fri Apr 24 18:27:24 MDT 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/applmathcomput2015.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0096300315003148",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied Mathematics and Computation",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00963003/",
}


bibtype: inproceedings
filename: elefunt.bib
label: deDinechin:2015:HIF
author: Florent de Dinechin and Matei Istoan
title: Hardware Implementations of Fixed-Point Atan2
crossref: Muller:2015:ISC
pages: 34--41
year: 2015
DOI: https://doi.org/10.1109/ARITH.2015.23
bibdate: Sat Aug 01 08:05:52 2015
bibtimestamp: 2015.08.01 08:05:52 ???
keywords: ARITH-22
acknowledgement: ack-nhfb
rawauthor: Florent de Dinechin and Matei Istoan
rawtitle: Hardware Implementations of Fixed-Point {Atan2}
entry:
@InProceedings{deDinechin:2015:HIF,
  author =       "Florent de Dinechin and Matei Istoan",
  title =        "Hardware Implementations of Fixed-Point {Atan2}",
  crossref =     "Muller:2015:ISC",
  pages =        "34--41",
  year =         "2015",
  DOI =          "https://doi.org/10.1109/ARITH.2015.23",
  bibdate =      "Sat Aug 01 08:05:52 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH-22",
}


bibtype: article
filename: elefunt.bib
label: Elezovic:2015:EPF
author: Neven Elezovic
title: Estimations of psi function and harmonic numbers
journal: j-APPL-MATH-COMP
volume: 258
pages: 192--205
day: 1
month: may
monthnumber: 5
year: 2015
CODEN: AMHCBQ
DOI: https://doi.org/10.1016/j.amc.2015.02.008
ISSN: 0096-3003 (print), 1873-5649 (electronic) OR 00963003 (print), 18735649 (electronic)
ISSNL: 0096-3003 OR 00963003
bibdate: Thu Mar 19 09:03:22 MDT 2015
bibtimestamp: 2015.03.19 09:03:22 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0096300315001617
fjournal: Applied Mathematics and Computation
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00963003/
rawauthor: Neven Elezovi{\'c}
rawtitle: Estimations of psi function and harmonic numbers
entry:
@Article{Elezovic:2015:EPF,
  author =       "Neven Elezovi{\'c}",
  title =        "Estimations of psi function and harmonic numbers",
  journal =      j-APPL-MATH-COMP,
  volume =       "258",
  number =       "??",
  pages =        "192--205",
  day =          "1",
  month =        may,
  year =         "2015",
  CODEN =        "AMHCBQ",
  DOI =          "https://doi.org/10.1016/j.amc.2015.02.008",
  ISSN =         "0096-3003 (print), 1873-5649 (electronic)",
  ISSN-L =       "0096-3003",
  bibdate =      "Thu Mar 19 09:03:22 MDT 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/applmathcomput2015.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0096300315001617",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied Mathematics and Computation",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00963003/",
}


bibtype: article
filename: elefunt.bib
label: Flocke:2015:AAE
author: N. Flocke
journal: j-TOMS
volume: 41
number: 4
pages: 30:1--30:24
month: oct
monthnumber: 10
year: 2015
CODEN: ACMSCU
DOI: https://doi.org/10.1145/2699468
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
bibdate: Mon Oct 26 17:31:15 MDT 2015
bibtimestamp: 2015.10.26 17:31:15 MDT
fjournal: ACM Transactions on Mathematical Software (TOMS)
acknowledgement: ack-nhfb
articleno: 30
journalURL: https://dl.acm.org/loi/toms
rawauthor: N. Flocke
title:
Algorithm 954: an Accurate and Efficient Cubic and Quartic Equation Solver for Physical Applications

bibsource:
http://www.acm.org/pubs/contents/journals/toms/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib

abstract:
We report on an accurate and efficient algorithm for obtaining all roots of general real cubic and quartic polynomials. Both the cubic and quartic solvers give highly accurate roots and place no restrictions on the magnitude of the polynomial coefficients. The key to the algorithm is a proper rescaling of both polynomials. This puts upper bounds on the magnitude of the roots and is very useful in stabilizing the root finding process. The cubic solver is based on dividing the cubic polynomial into six classes. By analyzing the root surface for each class, a fast convergent Newton--Raphson starting point for a real root is obtained at a cost no higher than three additions and four multiplications. The quartic solver uses the cubic solver in getting information about stationary points and, when the quartic has real roots, stable Newton--Raphson iterations give one of the extreme real roots. The remaining roots follow by composite deflation to a cubic. If the quartic has only complex roots, the present article shows that a stable Newton--Raphson iteration on a derived symmetric sixth degree polynomial can be formulated for the real parts of the complex roots. The imaginary parts follow by solving suitable quadratics.

rawabstract:
We report on an accurate and efficient algorithm for obtaining all roots of general real cubic and quartic polynomials. Both the cubic and quartic solvers give highly accurate roots and place no restrictions on the magnitude of the polynomial coefficients. The key to the algorithm is a proper rescaling of both polynomials. This puts upper bounds on the magnitude of the roots and is very useful in stabilizing the root finding process. The cubic solver is based on dividing the cubic polynomial into six classes. By analyzing the root surface for each class, a fast convergent Newton--Raphson starting point for a real root is obtained at a cost no higher than three additions and four multiplications. The quartic solver uses the cubic solver in getting information about stationary points and, when the quartic has real roots, stable Newton--Raphson iterations give one of the extreme real roots. The remaining roots follow by composite deflation to a cubic. If the quartic has only complex roots, the present article shows that a stable Newton--Raphson iteration on a derived symmetric sixth degree polynomial can be formulated for the real parts of the complex roots. The imaginary parts follow by solving suitable quadratics.

rawtitle:
{Algorithm 954}: an Accurate and Efficient Cubic and Quartic Equation Solver for Physical Applications

entry:
@Article{Flocke:2015:AAE,
  author =       "N. Flocke",
  title =        "{Algorithm 954}: an Accurate and Efficient Cubic and
                 Quartic Equation Solver for Physical Applications",
  journal =      j-TOMS,
  volume =       "41",
  number =       "4",
  pages =        "30:1--30:24",
  month =        oct,
  year =         "2015",
  CODEN =        "ACMSCU",
  DOI =          "https://doi.org/10.1145/2699468",
  ISSN =         "0098-3500 (print), 1557-7295 (electronic)",
  ISSN-L =       "0098-3500",
  bibdate =      "Mon Oct 26 17:31:15 MDT 2015",
  bibsource =    "http://www.acm.org/pubs/contents/journals/toms/;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/toms.bib",
  abstract =     "We report on an accurate and efficient algorithm for
                 obtaining all roots of general real cubic and quartic
                 polynomials. Both the cubic and quartic solvers give
                 highly accurate roots and place no restrictions on the
                 magnitude of the polynomial coefficients. The key to
                 the algorithm is a proper rescaling of both
                 polynomials. This puts upper bounds on the magnitude of
                 the roots and is very useful in stabilizing the root
                 finding process. The cubic solver is based on dividing
                 the cubic polynomial into six classes. By analyzing the
                 root surface for each class, a fast convergent
                 Newton--Raphson starting point for a real root is
                 obtained at a cost no higher than three additions and
                 four multiplications. The quartic solver uses the cubic
                 solver in getting information about stationary points
                 and, when the quartic has real roots, stable
                 Newton--Raphson iterations give one of the extreme real
                 roots. The remaining roots follow by composite
                 deflation to a cubic. If the quartic has only complex
                 roots, the present article shows that a stable
                 Newton--Raphson iteration on a derived symmetric sixth
                 degree polynomial can be formulated for the real parts
                 of the complex roots. The imaginary parts follow by
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  acknowledgement = ack-nhfb,
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  fjournal =     "ACM Transactions on Mathematical Software (TOMS)",
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}


bibtype: article
filename: elefunt.bib
label: Fukushima:2015:PFCa
author: Toshio Fukushima
journal: j-APPL-MATH-COMP
volume: 259
pages: 698--707
day: 15
month: may
monthnumber: 5
year: 2015
CODEN: AMHCBQ
ISSN: 0096-3003 (print), 1873-5649 (electronic) OR 00963003 (print), 18735649 (electronic)
ISSNL: 0096-3003 OR 00963003
bibdate: Fri Apr 24 18:27:24 MDT 2015
bibtimestamp: 2015.04.24 18:27:24 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0096300315003094
fjournal: Applied Mathematics and Computation
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00963003/
rawauthor: Toshio Fukushima
title:
Precise and fast computation of inverse Fermi--Dirac integral of order $ 1 / 2 $ by minimax rational function approximation

rawtitle:
Precise and fast computation of inverse {Fermi--Dirac} integral of order $ 1 / 2 $ by minimax rational function approximation

entry:
@Article{Fukushima:2015:PFCa,
  author =       "Toshio Fukushima",
  title =        "Precise and fast computation of inverse {Fermi--Dirac}
                 integral of order $ 1 / 2 $ by minimax rational
                 function approximation",
  journal =      j-APPL-MATH-COMP,
  volume =       "259",
  number =       "??",
  pages =        "698--707",
  day =          "15",
  month =        may,
  year =         "2015",
  CODEN =        "AMHCBQ",
  ISSN =         "0096-3003 (print), 1873-5649 (electronic)",
  ISSN-L =       "0096-3003",
  bibdate =      "Fri Apr 24 18:27:24 MDT 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/applmathcomput2015.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0096300315003094",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied Mathematics and Computation",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00963003/",
}


bibtype: article
filename: elefunt.bib
label: Fukushima:2015:PFCb
author: Toshio Fukushima
journal: j-APPL-MATH-COMP
volume: 259
pages: 708--729
day: 15
month: may
monthnumber: 5
year: 2015
CODEN: AMHCBQ
ISSN: 0096-3003 (print), 1873-5649 (electronic) OR 00963003 (print), 18735649 (electronic)
ISSNL: 0096-3003 OR 00963003
bibdate: Fri Apr 24 18:27:24 MDT 2015
bibtimestamp: 2015.04.24 18:27:24 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0096300315003033
fjournal: Applied Mathematics and Computation
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00963003/
rawauthor: Toshio Fukushima
title:
Precise and fast computation of Fermi--Dirac integral of integer and half integer order by piecewise minimax rational approximation

rawtitle:
Precise and fast computation of {Fermi--Dirac} integral of integer and half integer order by piecewise minimax rational approximation

entry:
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  title =        "Precise and fast computation of {Fermi--Dirac}
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  journal =      j-APPL-MATH-COMP,
  volume =       "259",
  number =       "??",
  pages =        "708--729",
  day =          "15",
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  bibdate =      "Fri Apr 24 18:27:24 MDT 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/applmathcomput2015.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0096300315003033",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied Mathematics and Computation",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00963003/",
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bibtype: inproceedings
filename: elefunt.bib
label: Fukushima:2015:PFCc
author: Toshio Fukushima
title: Precise and Fast Computation of Elliptic Integrals and Functions
crossref: Muller:2015:ISC
pages: 50--57
year: 2015
DOI: https://doi.org/10.1109/ARITH.2015.15
bibdate: Sat Aug 01 08:05:52 2015
bibtimestamp: 2015.08.01 08:05:52 ???
keywords: ARITH-22
acknowledgement: ack-nhfb
rawauthor: Toshio Fukushima
rawtitle: Precise and Fast Computation of Elliptic Integrals and Functions
entry:
@InProceedings{Fukushima:2015:PFCc,
  author =       "Toshio Fukushima",
  title =        "Precise and Fast Computation of Elliptic Integrals and
                 Functions",
  crossref =     "Muller:2015:ISC",
  pages =        "50--57",
  year =         "2015",
  DOI =          "https://doi.org/10.1109/ARITH.2015.15",
  bibdate =      "Sat Aug 01 08:05:52 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH-22",
}


bibtype: article
filename: elefunt.bib
label: Fukushima:2015:PFCd
author: Toshio Fukushima
journal: j-APPL-MATH-COMP
volume: 270
pages: 802--807
day: 1
month: nov
monthnumber: 11
year: 2015
CODEN: AMHCBQ
ISSN: 0096-3003 (print), 1873-5649 (electronic) OR 00963003 (print), 18735649 (electronic)
ISSNL: 0096-3003 OR 00963003
bibdate: Thu Nov 5 06:24:28 MST 2015
bibtimestamp: 2015.11.05 06:24:28 MST
URL: http://www.sciencedirect.com/science/article/pii/S0096300315011509
fjournal: Applied Mathematics and Computation
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00963003/
rawauthor: Toshio Fukushima
title:
Precise and fast computation of generalized Fermi--Dirac integral by parameter polynomial approximation

rawtitle:
Precise and fast computation of generalized {Fermi--Dirac} integral by parameter polynomial approximation

entry:
@Article{Fukushima:2015:PFCd,
  author =       "Toshio Fukushima",
  title =        "Precise and fast computation of generalized
                 {Fermi--Dirac} integral by parameter polynomial
                 approximation",
  journal =      j-APPL-MATH-COMP,
  volume =       "270",
  number =       "??",
  pages =        "802--807",
  day =          "1",
  month =        nov,
  year =         "2015",
  CODEN =        "AMHCBQ",
  ISSN =         "0096-3003 (print), 1873-5649 (electronic)",
  ISSN-L =       "0096-3003",
  bibdate =      "Thu Nov 5 06:24:28 MST 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/applmathcomput2015.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0096300315011509",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied Mathematics and Computation",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00963003/",
}


bibtype: article
filename: elefunt.bib
label: Gil:2015:CKF
author: Amparo Gil and Javier Segura and Nico M. Temme
title: Computing the Kummer function $ U(a, b, z) $ for small values of the arguments
journal: j-APPL-MATH-COMP
volume: 271
pages: 532--539
day: 15
month: nov
monthnumber: 11
year: 2015
CODEN: AMHCBQ
ISSN: 0096-3003 (print), 1873-5649 (electronic) OR 00963003 (print), 18735649 (electronic)
ISSNL: 0096-3003 OR 00963003
bibdate: Fri Nov 13 08:52:33 MST 2015
bibtimestamp: 2015.11.13 08:52:33 MST
URL: http://www.sciencedirect.com/science/article/pii/S0096300315012837
fjournal: Applied Mathematics and Computation
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00963003/
rawauthor: Amparo Gil and Javier Segura and Nico M. Temme
rawtitle: Computing the {Kummer} function {$ U(a, b, z) $} for small values of the arguments
entry:
@Article{Gil:2015:CKF,
  author =       "Amparo Gil and Javier Segura and Nico M. Temme",
  title =        "Computing the {Kummer} function {$ U(a, b, z) $} for
                 small values of the arguments",
  journal =      j-APPL-MATH-COMP,
  volume =       "271",
  number =       "??",
  pages =        "532--539",
  day =          "15",
  month =        nov,
  year =         "2015",
  CODEN =        "AMHCBQ",
  ISSN =         "0096-3003 (print), 1873-5649 (electronic)",
  ISSN-L =       "0096-3003",
  bibdate =      "Fri Nov 13 08:52:33 MST 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/applmathcomput2015.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0096300315012837",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied Mathematics and Computation",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00963003/",
}


bibtype: article
filename: elefunt.bib
label: Gil:2015:GPI
author: Amparo Gil and Javier Segura and Nico M. Temme
journal: j-COMP-PHYS-COMM
volume: 191
pages: 132--139
month: jun
monthnumber: 6
year: 2015
CODEN: CPHCBZ
ISSN: 0010-4655 (print), 1879-2944 (electronic) OR 00104655 (print), 18792944 (electronic)
ISSNL: 0010-4655 OR 00104655
bibdate: Fri Apr 24 18:44:55 MDT 2015
bibtimestamp: 2015.04.24 18:44:55 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0010465515000107
fjournal: Computer Physics Communications
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00104655/
rawauthor: Amparo Gil and Javier Segura and Nico M. Temme
title:
GammaCHI: a package for the inversion and computation of the gamma and chi-square cumulative distribution functions (central and noncentral)

rawtitle:
{GammaCHI}: a package for the inversion and computation of the gamma and chi-square cumulative distribution functions (central and noncentral)

entry:
@Article{Gil:2015:GPI,
  author =       "Amparo Gil and Javier Segura and Nico M. Temme",
  title =        "{GammaCHI}: a package for the inversion and
                 computation of the gamma and chi-square cumulative
                 distribution functions (central and noncentral)",
  journal =      j-COMP-PHYS-COMM,
  volume =       "191",
  number =       "??",
  pages =        "132--139",
  month =        jun,
  year =         "2015",
  CODEN =        "CPHCBZ",
  ISSN =         "0010-4655 (print), 1879-2944 (electronic)",
  ISSN-L =       "0010-4655",
  bibdate =      "Fri Apr 24 18:44:55 MDT 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/compphyscomm2010.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0010465515000107",
  acknowledgement = ack-nhfb,
  fjournal =     "Computer Physics Communications",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00104655/",
}


bibtype: article
filename: elefunt.bib
label: Graillat:2015:ECF
author: Stef Graillat and Christoph Lauter and Ping Tak Peter Tang and Naoya Yamanaka and Shin'ichi Oishi
title: Efficient Calculations of Faithfully Rounded $ l_2$-Norms of $n$-Vectors
journal: j-TOMS
volume: 41
number: 4
pages: 24:1--24:20
month: oct
monthnumber: 10
year: 2015
CODEN: ACMSCU
DOI: https://doi.org/10.1145/2699469
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
bibdate: Mon Oct 26 17:31:15 MDT 2015
bibtimestamp: 2015.10.26 17:31:15 MDT
fjournal: ACM Transactions on Mathematical Software (TOMS)
acknowledgement: ack-nhfb
articleno: 24
journalURL: https://dl.acm.org/loi/toms
rawauthor: Stef Graillat and Christoph Lauter and Ping Tak Peter Tang and Naoya Yamanaka and Shin'ichi Oishi
rawtitle: Efficient Calculations of Faithfully Rounded $ l_2$-Norms of $n$-Vectors
bibsource:
http://www.acm.org/pubs/contents/journals/toms/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib

abstract:
In this article, we present an efficient algorithm to compute the faithful rounding of the $ l_2 $-norm of a floating-point vector. This means that the result is accurate to within 1 bit of the underlying floating-point type. This algorithm does not generate overflows or underflows spuriously, but does so when the final result calls for such a numerical exception to be raised. Moreover, the algorithm is well suited for parallel implementation and vectorization. The implementation runs up to 3 times faster than the netlib version on current processors.

rawabstract:
In this article, we present an efficient algorithm to compute the faithful rounding of the $ l_2 $-norm of a floating-point vector. This means that the result is accurate to within 1 bit of the underlying floating-point type. This algorithm does not generate overflows or underflows spuriously, but does so when the final result calls for such a numerical exception to be raised. Moreover, the algorithm is well suited for parallel implementation and vectorization. The implementation runs up to 3 times faster than the netlib version on current processors.

entry:
@Article{Graillat:2015:ECF,
  author =       "Stef Graillat and Christoph Lauter and Ping Tak Peter
                 Tang and Naoya Yamanaka and Shin'ichi Oishi",
  title =        "Efficient Calculations of Faithfully Rounded $
                 l_2$-Norms of $n$-Vectors",
  journal =      j-TOMS,
  volume =       "41",
  number =       "4",
  pages =        "24:1--24:20",
  month =        oct,
  year =         "2015",
  CODEN =        "ACMSCU",
  DOI =          "https://doi.org/10.1145/2699469",
  ISSN =         "0098-3500 (print), 1557-7295 (electronic)",
  ISSN-L =       "0098-3500",
  bibdate =      "Mon Oct 26 17:31:15 MDT 2015",
  bibsource =    "http://www.acm.org/pubs/contents/journals/toms/;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/toms.bib",
  abstract =     "In this article, we present an efficient algorithm to
                 compute the faithful rounding of the $ l_2 $-norm of a
                 floating-point vector. This means that the result is
                 accurate to within 1 bit of the underlying
                 floating-point type. This algorithm does not generate
                 overflows or underflows spuriously, but does so when
                 the final result calls for such a numerical exception
                 to be raised. Moreover, the algorithm is well suited
                 for parallel implementation and vectorization. The
                 implementation runs up to 3 times faster than the
                 netlib version on current processors.",
  acknowledgement = ack-nhfb,
  articleno =    "24",
  fjournal =     "ACM Transactions on Mathematical Software (TOMS)",
  journal-URL =  "https://dl.acm.org/loi/toms",
}


bibtype: article
filename: elefunt.bib
label: Graillat:2015:MRE
author: Stef Graillat and Vincent Lefevre and Jean-Michel Muller
journal: j-NUMER-ALGORITHMS
volume: 70
number: 3
pages: 653--667
month: nov
monthnumber: 11
year: 2015
CODEN: NUALEG
DOI: https://doi.org/10.1007/s11075-015-9967-8
ISSN: 1017-1398 (print), 1572-9265 (electronic) OR 10171398 (print), 15729265 (electronic)
ISSNL: 1017-1398 OR 10171398
bibdate: Sun Oct 25 07:27:50 MDT 2015
bibtimestamp: 2015.10.25 07:27:50 MDT
URL: http://link.springer.com/article/10.1007/s11075-015-9967-8
fjournal: Numerical Algorithms
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11075
rawauthor: Stef Graillat and Vincent Lef{\`e}vre and Jean-Michel Muller
title:
On the maximum relative error when computing integer powers by iterated multiplications in floating-point arithmetic

bibsource:
http://link.springer.com/journal/11075/70/3; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib

remark:
The authors show via a complex multipage proof that the iterated product for $ x^n $ in p-bit binary arithmetic with default IEEE 754 rounding (to nearest with ties to even) produces a worst-case relative error in the product that is no larger than $ (n - 1) u $, where $ u = 2^-p $ is the rounding unit.

rawtitle:
On the maximum relative error when computing integer powers by iterated multiplications in floating-point arithmetic

entry:
@Article{Graillat:2015:MRE,
  author =       "Stef Graillat and Vincent Lef{\`e}vre and Jean-Michel
                 Muller",
  title =        "On the maximum relative error when computing integer
                 powers by iterated multiplications in floating-point
                 arithmetic",
  journal =      j-NUMER-ALGORITHMS,
  volume =       "70",
  number =       "3",
  pages =        "653--667",
  month =        nov,
  year =         "2015",
  CODEN =        "NUALEG",
  DOI =          "https://doi.org/10.1007/s11075-015-9967-8",
  ISSN =         "1017-1398 (print), 1572-9265 (electronic)",
  ISSN-L =       "1017-1398",
  bibdate =      "Sun Oct 25 07:27:50 MDT 2015",
  bibsource =    "http://link.springer.com/journal/11075/70/3;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib",
  URL =          "http://link.springer.com/article/10.1007/s11075-015-9967-8",
  acknowledgement = ack-nhfb,
  fjournal =     "Numerical Algorithms",
  journal-URL =  "http://link.springer.com/journal/11075",
  remark =       "The authors show via a complex multipage proof that
                 the iterated product for $ x^n $ in p-bit binary
                 arithmetic with default IEEE 754 rounding (to nearest
                 with ties to even) produces a worst-case relative error
                 in the product that is no larger than $ (n - 1) u $,
                 where $ u = 2^{-p} $ is the rounding unit.",
}


bibtype: inproceedings
filename: elefunt.bib
label: Johansson:2015:EIE
author: Fredrik Johansson
title: Efficient Implementation of Elementary Functions in the Medium-Precision Range
crossref: Muller:2015:ISC
pages: 83--89
year: 2015
DOI: https://doi.org/10.1109/ARITH.2015.16
bibdate: Sat Aug 01 08:05:52 2015
bibtimestamp: 2015.08.01 08:05:52 ???
keywords: ARITH-22
acknowledgement: ack-nhfb
rawauthor: Fredrik Johansson
rawtitle: Efficient Implementation of Elementary Functions in the Medium-Precision Range
entry:
@InProceedings{Johansson:2015:EIE,
  author =       "Fredrik Johansson",
  title =        "Efficient Implementation of Elementary Functions in
                 the Medium-Precision Range",
  crossref =     "Muller:2015:ISC",
  pages =        "83--89",
  year =         "2015",
  DOI =          "https://doi.org/10.1109/ARITH.2015.16",
  bibdate =      "Sat Aug 01 08:05:52 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH-22",
}


bibtype: article
filename: elefunt.bib
label: Johansson:2015:RHP
author: Fredrik Johansson
title: Rigorous high-precision computation of the Hurwitz zeta function and its derivatives
journal: j-NUMER-ALGORITHMS
volume: 69
number: 2
pages: 253--270
month: jun
monthnumber: 6
year: 2015
CODEN: NUALEG
DOI: https://doi.org/10.1007/s11075-014-9893-1
ISSN: 1017-1398 (print), 1572-9265 (electronic) OR 10171398 (print), 15729265 (electronic)
ISSNL: 1017-1398 OR 10171398
bibdate: Thu May 28 15:00:06 MDT 2015
bibtimestamp: 2015.05.28 15:00:06 MDT
URL: http://link.springer.com/article/10.1007/s11075-014-9893-1
fjournal: Numerical Algorithms
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11075
rawauthor: Fredrik Johansson
rawtitle: Rigorous high-precision computation of the {Hurwitz} zeta function and its derivatives
bibsource:
http://link.springer.com/journal/11075/69/2; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib

entry:
@Article{Johansson:2015:RHP,
  author =       "Fredrik Johansson",
  title =        "Rigorous high-precision computation of the {Hurwitz}
                 zeta function and its derivatives",
  journal =      j-NUMER-ALGORITHMS,
  volume =       "69",
  number =       "2",
  pages =        "253--270",
  month =        jun,
  year =         "2015",
  CODEN =        "NUALEG",
  DOI =          "https://doi.org/10.1007/s11075-014-9893-1",
  ISSN =         "1017-1398 (print), 1572-9265 (electronic)",
  ISSN-L =       "1017-1398",
  bibdate =      "Thu May 28 15:00:06 MDT 2015",
  bibsource =    "http://link.springer.com/journal/11075/69/2;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib",
  URL =          "http://link.springer.com/article/10.1007/s11075-014-9893-1",
  acknowledgement = ack-nhfb,
  fjournal =     "Numerical Algorithms",
  journal-URL =  "http://link.springer.com/journal/11075",
}


bibtype: article
filename: elefunt.bib
label: Kuznetsov:2015:CTT
author: A. Kuznetsov
title: Computing the truncated theta function via Mordell integral
journal: j-MATH-COMPUT
volume: 84
number: 296
pages: 2911--2926
year: 2015
CODEN: MCMPAF
ISSN: 0025-5718 (print), 1088-6842 (electronic) OR 00255718 (print), 10886842 (electronic)
ISSNL: 0025-5718 OR 00255718
bibdate: Tue Oct 20 16:30:35 MDT 2015
bibtimestamp: 2015.10.20 16:30:35 MDT
fjournal: Mathematics of Computation
acknowledgement: ack-nhfb
journalURL: http://www.ams.org/mcom/
rawauthor: A. Kuznetsov
rawtitle: Computing the truncated theta function via {Mordell} integral
bibsource:
http://www.ams.org/mcom/2015-84-296; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp2010.bib

URL:
http://www.ams.org/journals/mcom/2015-84-296/S0025-5718-2015-02953-6; http://www.ams.org/journals/mcom/2015-84-296/S0025-5718-2015-02953-6/S0025-5718-2015-02953-6.pdf

entry:
@Article{Kuznetsov:2015:CTT,
  author =       "A. Kuznetsov",
  title =        "Computing the truncated theta function via {Mordell}
                 integral",
  journal =      j-MATH-COMPUT,
  volume =       "84",
  number =       "296",
  pages =        "2911--2926",
  month =        "",
  year =         "2015",
  CODEN =        "MCMPAF",
  ISSN =         "0025-5718 (print), 1088-6842 (electronic)",
  ISSN-L =       "0025-5718",
  bibdate =      "Tue Oct 20 16:30:35 MDT 2015",
  bibsource =    "http://www.ams.org/mcom/2015-84-296;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/mathcomp2010.bib",
  URL =          "http://www.ams.org/journals/mcom/2015-84-296/S0025-5718-2015-02953-6;
                 http://www.ams.org/journals/mcom/2015-84-296/S0025-5718-2015-02953-6/S0025-5718-2015-02953-6.pdf",
  acknowledgement = ack-nhfb,
  fjournal =     "Mathematics of Computation",
  journal-URL =  "http://www.ams.org/mcom/",
}


bibtype: inproceedings
filename: elefunt.bib
label: Lauter:2015:SAF
author: Christoph Lauter and Marc Mezzarobba
title: Semi-Automatic Floating-Point Implementation of Special Functions
crossref: Muller:2015:ISC
pages: 58--65
year: 2015
DOI: https://doi.org/10.1109/ARITH.2015.12
bibdate: Sat Aug 01 08:05:52 2015
bibtimestamp: 2015.08.01 08:05:52 ???
keywords: ARITH-22
acknowledgement: ack-nhfb
rawauthor: Christoph Lauter and Marc Mezzarobba
rawtitle: Semi-Automatic Floating-Point Implementation of Special Functions
entry:
@InProceedings{Lauter:2015:SAF,
  author =       "Christoph Lauter and Marc Mezzarobba",
  title =        "Semi-Automatic Floating-Point Implementation of
                 Special Functions",
  crossref =     "Muller:2015:ISC",
  pages =        "58--65",
  year =         "2015",
  DOI =          "https://doi.org/10.1109/ARITH.2015.12",
  bibdate =      "Sat Aug 01 08:05:52 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH-22",
}


bibtype: article
filename: elefunt.bib
label: Lu:2015:NSC
author: Dawei Lu and Lixin Song and Yang Yu
title: New sequences with continued fraction towards Euler's constant
journal: j-APPL-MATH-COMP
volume: 259
pages: 12--20
day: 15
month: may
monthnumber: 5
year: 2015
CODEN: AMHCBQ
ISSN: 0096-3003 (print), 1873-5649 (electronic) OR 00963003 (print), 18735649 (electronic)
ISSNL: 0096-3003 OR 00963003
bibdate: Fri Apr 24 18:27:24 MDT 2015
bibtimestamp: 2015.04.24 18:27:24 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0096300315001745
fjournal: Applied Mathematics and Computation
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00963003/
rawauthor: Dawei Lu and Lixin Song and Yang Yu
rawtitle: New sequences with continued fraction towards {Euler}'s constant
entry:
@Article{Lu:2015:NSC,
  author =       "Dawei Lu and Lixin Song and Yang Yu",
  title =        "New sequences with continued fraction towards
                 {Euler}'s constant",
  journal =      j-APPL-MATH-COMP,
  volume =       "259",
  number =       "??",
  pages =        "12--20",
  day =          "15",
  month =        may,
  year =         "2015",
  CODEN =        "AMHCBQ",
  ISSN =         "0096-3003 (print), 1873-5649 (electronic)",
  ISSN-L =       "0096-3003",
  bibdate =      "Fri Apr 24 18:27:24 MDT 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/applmathcomput2015.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0096300315001745",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied Mathematics and Computation",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00963003/",
}


bibtype: article
filename: elefunt.bib
label: Lu:2015:SNA
author: Dawei Lu and Lixin Song and Congxu Ma
journal: j-APPL-MATH-COMP
volume: 253
pages: 1--7
day: 15
month: feb
monthnumber: 2
year: 2015
CODEN: AMHCBQ
DOI: https://doi.org/10.1016/j.amc.2014.12.077
ISSN: 0096-3003 (print), 1873-5649 (electronic) OR 00963003 (print), 18735649 (electronic)
ISSNL: 0096-3003 OR 00963003
bibdate: Wed Feb 18 09:36:23 MST 2015
bibtimestamp: 2015.02.18 09:36:23 MST
URL: http://www.sciencedirect.com/science/article/pii/S0096300314017317
fjournal: Applied Mathematics and Computation
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00963003/
rawauthor: Dawei Lu and Lixin Song and Congxu Ma
title:
Some new asymptotic approximations of the gamma function based on Nemes' formula, Ramanujan's formula and Burnside's formula

rawtitle:
Some new asymptotic approximations of the gamma function based on {Nemes}' formula, {Ramanujan}'s formula and {Burnside}'s formula

entry:
@Article{Lu:2015:SNA,
  author =       "Dawei Lu and Lixin Song and Congxu Ma",
  title =        "Some new asymptotic approximations of the gamma
                 function based on {Nemes}' formula, {Ramanujan}'s
                 formula and {Burnside}'s formula",
  journal =      j-APPL-MATH-COMP,
  volume =       "253",
  number =       "??",
  pages =        "1--7",
  day =          "15",
  month =        feb,
  year =         "2015",
  CODEN =        "AMHCBQ",
  DOI =          "https://doi.org/10.1016/j.amc.2014.12.077",
  ISSN =         "0096-3003 (print), 1873-5649 (electronic)",
  ISSN-L =       "0096-3003",
  bibdate =      "Wed Feb 18 09:36:23 MST 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/applmathcomput2015.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0096300314017317",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied Mathematics and Computation",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00963003/",
}


bibtype: article
filename: elefunt.bib
label: Lu:2015:SNQ
author: Dawei Lu and Congxu Ma
journal: j-NUMER-ALGORITHMS
volume: 70
number: 4
pages: 825--833
month: dec
monthnumber: 12
year: 2015
CODEN: NUALEG
DOI: https://doi.org/10.1007/s11075-015-9975-8
ISSN: 1017-1398 (print), 1572-9265 (electronic) OR 10171398 (print), 15729265 (electronic)
ISSNL: 1017-1398 OR 10171398
bibdate: Mon Jan 25 08:55:03 MST 2016
bibtimestamp: 2016.01.25 08:55:03 MST
URL: http://link.springer.com/article/10.1007/s11075-015-9975-8
fjournal: Numerical Algorithms
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11075
rawauthor: Dawei Lu and Congxu Ma
title:
Some new quicker continued fraction approximations for the gamma function related to the Nemes' formula

bibsource:
http://link.springer.com/journal/11075/70/4; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib

rawtitle:
Some new quicker continued fraction approximations for the gamma function related to the {Nemes}' formula

entry:
@Article{Lu:2015:SNQ,
  author =       "Dawei Lu and Congxu Ma",
  title =        "Some new quicker continued fraction approximations for
                 the gamma function related to the {Nemes}' formula",
  journal =      j-NUMER-ALGORITHMS,
  volume =       "70",
  number =       "4",
  pages =        "825--833",
  month =        dec,
  year =         "2015",
  CODEN =        "NUALEG",
  DOI =          "https://doi.org/10.1007/s11075-015-9975-8",
  ISSN =         "1017-1398 (print), 1572-9265 (electronic)",
  ISSN-L =       "1017-1398",
  bibdate =      "Mon Jan 25 08:55:03 MST 2016",
  bibsource =    "http://link.springer.com/journal/11075/70/4;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib",
  URL =          "http://link.springer.com/article/10.1007/s11075-015-9975-8",
  acknowledgement = ack-nhfb,
  fjournal =     "Numerical Algorithms",
  journal-URL =  "http://link.springer.com/journal/11075",
}


bibtype: article
filename: elefunt.bib
label: Mortici:2015:PAG
author: Cristinel Mortici and Hari M. Srivastava
title: A product approximation of the gamma function
journal: j-NUMER-ALGORITHMS
volume: 69
number: 3
pages: 595--610
month: jul
monthnumber: 7
year: 2015
CODEN: NUALEG
DOI: https://doi.org/10.1007/s11075-014-9915-z
ISSN: 1017-1398 (print), 1572-9265 (electronic) OR 10171398 (print), 15729265 (electronic)
ISSNL: 1017-1398 OR 10171398
bibdate: Sat Aug 8 13:58:48 MDT 2015
bibtimestamp: 2015.08.08 13:58:48 MDT
URL: http://link.springer.com/article/10.1007/s11075-014-9915-z
fjournal: Numerical Algorithms
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11075
rawauthor: Cristinel Mortici and Hari M. Srivastava
rawtitle: A product approximation of the gamma function
bibsource:
http://link.springer.com/journal/11075/69/3; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib

entry:
@Article{Mortici:2015:PAG,
  author =       "Cristinel Mortici and Hari M. Srivastava",
  title =        "A product approximation of the gamma function",
  journal =      j-NUMER-ALGORITHMS,
  volume =       "69",
  number =       "3",
  pages =        "595--610",
  month =        jul,
  year =         "2015",
  CODEN =        "NUALEG",
  DOI =          "https://doi.org/10.1007/s11075-014-9915-z",
  ISSN =         "1017-1398 (print), 1572-9265 (electronic)",
  ISSN-L =       "1017-1398",
  bibdate =      "Sat Aug 8 13:58:48 MDT 2015",
  bibsource =    "http://link.springer.com/journal/11075/69/3;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib",
  URL =          "http://link.springer.com/article/10.1007/s11075-014-9915-z",
  acknowledgement = ack-nhfb,
  fjournal =     "Numerical Algorithms",
  journal-URL =  "http://link.springer.com/journal/11075",
}


bibtype: article
filename: elefunt.bib
label: Nadarajah:2015:CGH
author: Saralees Nadarajah
title: On the Computation of Gauss Hypergeometric Functions
journal: j-AMER-STAT
volume: 69
number: 2
pages: 146--148
year: 2015
CODEN: ASTAAJ
DOI: https://doi.org/10.1080/00031305.2015.1028595
ISSN: 0003-1305 (print), 1537-2731 (electronic) OR 00031305 (print), 15372731 (electronic)
ISSNL: 0003-1305 OR 00031305
bibdate: Sun Aug 9 16:54:48 MDT 2015
bibtimestamp: 2015.08.09 16:54:48 MDT
fjournal: The American Statistician
acknowledgement: ack-nhfb
journalURL: http://amstat.tandfonline.com/loi/utas20
onlinedate: 24 Mar 2015
rawauthor: Saralees Nadarajah
rawtitle: On the Computation of {Gauss} Hypergeometric Functions
bibsource:
http://www.tandfonline.com/toc/utas20/69/2; https://www.math.utah.edu/pub/tex/bib/amstat2010.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

entry:
@Article{Nadarajah:2015:CGH,
  author =       "Saralees Nadarajah",
  title =        "On the Computation of {Gauss} Hypergeometric
                 Functions",
  journal =      j-AMER-STAT,
  volume =       "69",
  number =       "2",
  pages =        "146--148",
  year =         "2015",
  CODEN =        "ASTAAJ",
  DOI =          "https://doi.org/10.1080/00031305.2015.1028595",
  ISSN =         "0003-1305 (print), 1537-2731 (electronic)",
  ISSN-L =       "0003-1305",
  bibdate =      "Sun Aug 9 16:54:48 MDT 2015",
  bibsource =    "http://www.tandfonline.com/toc/utas20/69/2;
                 https://www.math.utah.edu/pub/tex/bib/amstat2010.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "The American Statistician",
  journal-URL =  "http://amstat.tandfonline.com/loi/utas20",
  onlinedate =   "24 Mar 2015",
}


bibtype: article
filename: elefunt.bib
label: Natalini:2015:BPM
author: Pierpaolo Natalini and Paolo Emilio Ricci
title: Bell polynomials and modified Bessel functions of half-integral order
journal: j-APPL-MATH-COMP
volume: 268
pages: 270--274
day: 1
month: oct
monthnumber: 10
year: 2015
CODEN: AMHCBQ
DOI: https://doi.org/10.1016/j.amc.2015.06.069
ISSN: 0096-3003 (print), 1873-5649 (electronic) OR 00963003 (print), 18735649 (electronic)
ISSNL: 0096-3003 OR 00963003
bibdate: Wed Sep 16 06:56:32 MDT 2015
bibtimestamp: 2015.09.16 06:56:32 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0096300315008504
fjournal: Applied Mathematics and Computation
keywords: Bell polynomials; Bessel functions; Combinatorial analysis; Hankel functions
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00963003/
rawauthor: Pierpaolo Natalini and Paolo Emilio Ricci
rawtitle: {Bell} polynomials and modified {Bessel} functions of half-integral order
entry:
@Article{Natalini:2015:BPM,
  author =       "Pierpaolo Natalini and Paolo Emilio Ricci",
  title =        "{Bell} polynomials and modified {Bessel} functions of
                 half-integral order",
  journal =      j-APPL-MATH-COMP,
  volume =       "268",
  number =       "??",
  pages =        "270--274",
  day =          "1",
  month =        oct,
  year =         "2015",
  CODEN =        "AMHCBQ",
  DOI =          "https://doi.org/10.1016/j.amc.2015.06.069",
  ISSN =         "0096-3003 (print), 1873-5649 (electronic)",
  ISSN-L =       "0096-3003",
  bibdate =      "Wed Sep 16 06:56:32 MDT 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/applmathcomput2015.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0096300315008504",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied Mathematics and Computation",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00963003/",
  keywords =     "Bell polynomials; Bessel functions; Combinatorial
                 analysis; Hankel functions",
}


bibtype: article
filename: elefunt.bib
label: Qi:2015:SIT
author: Feng Qi and Cristinel Mortici
title: Some inequalities for the trigamma function in terms of the digamma function
journal: j-APPL-MATH-COMP
volume: 271
pages: 502--511
day: 15
month: nov
monthnumber: 11
year: 2015
CODEN: AMHCBQ
ISSN: 0096-3003 (print), 1873-5649 (electronic) OR 00963003 (print), 18735649 (electronic)
ISSNL: 0096-3003 OR 00963003
bibdate: Fri Nov 13 08:52:33 MST 2015
bibtimestamp: 2015.11.13 08:52:33 MST
URL: http://www.sciencedirect.com/science/article/pii/S0096300315012758
fjournal: Applied Mathematics and Computation
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00963003/
rawauthor: Feng Qi and Cristinel Mortici
rawtitle: Some inequalities for the trigamma function in terms of the digamma function
entry:
@Article{Qi:2015:SIT,
  author =       "Feng Qi and Cristinel Mortici",
  title =        "Some inequalities for the trigamma function in terms
                 of the digamma function",
  journal =      j-APPL-MATH-COMP,
  volume =       "271",
  number =       "??",
  pages =        "502--511",
  day =          "15",
  month =        nov,
  year =         "2015",
  CODEN =        "AMHCBQ",
  ISSN =         "0096-3003 (print), 1873-5649 (electronic)",
  ISSN-L =       "0096-3003",
  bibdate =      "Fri Nov 13 08:52:33 MST 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/applmathcomput2015.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0096300315012758",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied Mathematics and Computation",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00963003/",
}


bibtype: book
filename: elefunt.bib
label: Schwalm:2015:EFE
author: William A. Schwalm
title: Elliptic Functions and Elliptic Integrals
publisher: Morgan and Claypool Publishers and IOP Publishing
address: San Rafael, CA, USA and Bristol, UK
pages: 67
year: 2015
DOI: https://doi.org/10.1088/978-1-6817-4230-4
ISSN: 2054-7307 OR 20547307
LCCN: QA343 .S355 2015
bibdate: Tue Mar 14 07:38:46 MDT 2023
bibsource: fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2023.03.14 07:38:46 MDT
series: IOP concise physics
URL: http://iopscience.iop.org/book/978-1-6817-4230-4
acknowledgement: ack-nhfb
rawauthor: William A. Schwalm
rawtitle: Elliptic Functions and Elliptic Integrals
ISBN:
1-68174-166-0 (print), 1-68174-230-6 (e-book), 1-68174-102-4 (mobi) OR 1681741660 (print), 1681742306 (ebook), 1681741024 (mobi)

ISBN13:
978-1-68174-166-6 (print), 978-1-68174-230-4 (e-book), 978-1-68174-102-4 (mobi) OR 9781681741666 (print), 9781681742304 (ebook), 9781681741024 (mobi)

abstract:
This volume is a basic introduction to certain aspects of elliptic functions and elliptic integrals. Primarily, the elliptic functions stand out as closed solutions to a class of physical and geometrical problems giving rise to nonlinear differential equations. While these nonlinear equations may not be the types of greatest interest currently, the fact that they are solvable exactly in terms of functions about which much is known makes up for this. The elliptic functions of Jacobi, or equivalently the Weierstrass elliptic functions, inhabit the literature on current problems in condensed matter and statistical physics, on solitons and conformal representations, and all sorts of famous problems in classical mechanics. The lectures on elliptic functions have evolved as part of the first semester of a course on theoretical and mathematical methods given to first- and second-year graduate students in physics and chemistry at the University of North Dakota. They are for graduate students or for researchers who want an elementary introduction to the subject that nevertheless leaves them with enough of the details to address real problems. The style is supposed to be informal. The intention is to introduce the subject as a moderate extension of ordinary trigonometry in which the reference circle is replaced by an ellipse. This entre depends upon fewer tools and has seemed less intimidating that other typical introductions to the subject that depend on some knowledge of complex variables. The first three lectures assume only calculus, including the chain rule and elementary knowledge of differential equations. In the later lectures, the complex analytic properties are introduced naturally so that a more complete study becomes possible

TOC:
Preface 1. Elliptic functions as trigonometry 1.1. Definition of Jacobian elliptic functions and trigonometric identities 1.2. Differential equations 1.3. Anharmonic oscillator 2. Differential equations satisfied by the Jacobi elliptic functions: pendula 2.1. Oscillatory motion of a pendulum at large amplitude 2.2. Motion traversing the whole circle 2.3. The sine-Gordon equation: a series of pendula 2.4. Series of pendula: 'super luminal' case 3. General reduction of the DE in terms of Jacobi functions 3.1. Linear fractional transformation and cross ratio 3.2. Reduction of general quartic case 3.3. Finding the coefficients of the linear fractional transformation 4. Elliptic integrals 4.1. Review of complex variables up through residues 4.2. Branching and multi-valued functions in complex planes 4.3. Elliptic integrals and elliptic functions in complex planes 4.4. Example 4.5. Reduction of the most general elliptic integral in terms of the three Legendre forms

rawabstract:
This volume is a basic introduction to certain aspects of elliptic functions and elliptic integrals. Primarily, the elliptic functions stand out as closed solutions to a class of physical and geometrical problems giving rise to nonlinear differential equations. While these nonlinear equations may not be the types of greatest interest currently, the fact that they are solvable exactly in terms of functions about which much is known makes up for this. The elliptic functions of Jacobi, or equivalently the Weierstrass elliptic functions, inhabit the literature on current problems in condensed matter and statistical physics, on solitons and conformal representations, and all sorts of famous problems in classical mechanics. The lectures on elliptic functions have evolved as part of the first semester of a course on theoretical and mathematical methods given to first- and second-year graduate students in physics and chemistry at the University of North Dakota. They are for graduate students or for researchers who want an elementary introduction to the subject that nevertheless leaves them with enough of the details to address real problems. The style is supposed to be informal. The intention is to introduce the subject as a moderate extension of ordinary trigonometry in which the reference circle is replaced by an ellipse. This entre depends upon fewer tools and has seemed less intimidating that other typical introductions to the subject that depend on some knowledge of complex variables. The first three lectures assume only calculus, including the chain rule and elementary knowledge of differential equations. In the later lectures, the complex analytic properties are introduced naturally so that a more complete study becomes possible

rawTOC:
Preface \\ 1. Elliptic functions as trigonometry \\ 1.1. Definition of Jacobian elliptic functions and trigonometric identities \\ 1.2. Differential equations \\ 1.3. Anharmonic oscillator \\ 2. Differential equations satisfied by the Jacobi elliptic functions: pendula \\ 2.1. Oscillatory motion of a pendulum at large amplitude \\ 2.2. Motion traversing the whole circle \\ 2.3. The sine-Gordon equation: a series of pendula \\ 2.4. Series of pendula: 'super luminal' case \\ 3. General reduction of the DE in terms of Jacobi functions \\ 3.1. Linear fractional transformation and cross ratio \\ 3.2. Reduction of general quartic case \\ 3.3. Finding the coefficients of the linear fractional transformation \\ 4. Elliptic integrals \\ 4.1. Review of complex variables up through residues \\ 4.2. Branching and multi-valued functions in complex planes \\ 4.3. Elliptic integrals and elliptic functions in complex planes \\ 4.4. Example \\ 4.5. Reduction of the most general elliptic integral in terms of the three Legendre forms

entry:
@Book{Schwalm:2015:EFE,
  author =       "William A. Schwalm",
  title =        "Elliptic Functions and Elliptic Integrals",
  publisher =    "Morgan and Claypool Publishers and IOP Publishing",
  address =      "San Rafael, CA, USA and Bristol, UK",
  pages =        "67",
  year =         "2015",
  DOI =          "https://doi.org/10.1088/978-1-6817-4230-4",
  ISBN =         "1-68174-166-0 (print), 1-68174-230-6 (e-book),
                 1-68174-102-4 (mobi)",
  ISBN-13 =      "978-1-68174-166-6 (print), 978-1-68174-230-4 (e-book),
                 978-1-68174-102-4 (mobi)",
  ISSN =         "2054-7307",
  LCCN =         "QA343 .S355 2015",
  bibdate =      "Tue Mar 14 07:38:46 MDT 2023",
  bibsource =    "fsz3950.oclc.org:210/WorldCat;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  series =       "IOP concise physics",
  URL =          "http://iopscience.iop.org/book/978-1-6817-4230-4",
  abstract =     "This volume is a basic introduction to certain aspects
                 of elliptic functions and elliptic integrals.
                 Primarily, the elliptic functions stand out as closed
                 solutions to a class of physical and geometrical
                 problems giving rise to nonlinear differential
                 equations. While these nonlinear equations may not be
                 the types of greatest interest currently, the fact that
                 they are solvable exactly in terms of functions about
                 which much is known makes up for this. The elliptic
                 functions of Jacobi, or equivalently the Weierstrass
                 elliptic functions, inhabit the literature on current
                 problems in condensed matter and statistical physics,
                 on solitons and conformal representations, and all
                 sorts of famous problems in classical mechanics. The
                 lectures on elliptic functions have evolved as part of
                 the first semester of a course on theoretical and
                 mathematical methods given to first- and second-year
                 graduate students in physics and chemistry at the
                 University of North Dakota. They are for graduate
                 students or for researchers who want an elementary
                 introduction to the subject that nevertheless leaves
                 them with enough of the details to address real
                 problems. The style is supposed to be informal. The
                 intention is to introduce the subject as a moderate
                 extension of ordinary trigonometry in which the
                 reference circle is replaced by an ellipse. This entre
                 depends upon fewer tools and has seemed less
                 intimidating that other typical introductions to the
                 subject that depend on some knowledge of complex
                 variables. The first three lectures assume only
                 calculus, including the chain rule and elementary
                 knowledge of differential equations. In the later
                 lectures, the complex analytic properties are
                 introduced naturally so that a more complete study
                 becomes possible",
  acknowledgement = ack-nhfb,
  tableofcontents = "Preface \\
                 1. Elliptic functions as trigonometry \\
                 1.1. Definition of Jacobian elliptic functions and
                 trigonometric identities \\
                 1.2. Differential equations \\
                 1.3. Anharmonic oscillator \\
                 2. Differential equations satisfied by the Jacobi
                 elliptic functions: pendula \\
                 2.1. Oscillatory motion of a pendulum at large
                 amplitude \\
                 2.2. Motion traversing the whole circle \\
                 2.3. The sine-Gordon equation: a series of pendula \\
                 2.4. Series of pendula: 'super luminal' case \\
                 3. General reduction of the DE in terms of Jacobi
                 functions \\
                 3.1. Linear fractional transformation and cross ratio
                 \\
                 3.2. Reduction of general quartic case \\
                 3.3. Finding the coefficients of the linear fractional
                 transformation \\
                 4. Elliptic integrals \\
                 4.1. Review of complex variables up through residues
                 \\
                 4.2. Branching and multi-valued functions in complex
                 planes \\
                 4.3. Elliptic integrals and elliptic functions in
                 complex planes \\
                 4.4. Example \\
                 4.5. Reduction of the most general elliptic integral in
                 terms of the three Legendre forms",
}


bibtype: article
filename: elefunt.bib
label: Sun:2015:LEG
author: Qiming Sun
title: Libcint: an efficient general integral library for Gaussian basis functions
journal: j-J-COMPUT-CHEM
volume: 36
number: 22
pages: 1664--1671
day: 15
month: aug
monthnumber: 8
year: 2015
CODEN: JCCHDD
DOI: https://doi.org/10.1002/jcc.23981
ISSN: 0192-8651 (print), 1096-987X (electronic) OR 01928651 (print), 1096987X (electronic)
ISSNL: 0192-8651 OR 01928651
bibdate: Sat Jul 25 20:32:36 MDT 2015
bibtimestamp: 2015.07.25 20:32:36 MDT
fjournal: Journal of Computational Chemistry
acknowledgement: ack-nhfb
journalURL: http://www.interscience.wiley.com/jpages/0192-8651
onlinedate: 30 Jun 2015
rawauthor: Qiming Sun
rawtitle: {Libcint}: an efficient general integral library for {Gaussian} basis functions
bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputchem2010.bib

entry:
@Article{Sun:2015:LEG,
  author =       "Qiming Sun",
  title =        "{Libcint}: an efficient general integral library for
                 {Gaussian} basis functions",
  journal =      j-J-COMPUT-CHEM,
  volume =       "36",
  number =       "22",
  pages =        "1664--1671",
  day =          "15",
  month =        aug,
  year =         "2015",
  CODEN =        "JCCHDD",
  DOI =          "https://doi.org/10.1002/jcc.23981",
  ISSN =         "0192-8651 (print), 1096-987X (electronic)",
  ISSN-L =       "0192-8651",
  bibdate =      "Sat Jul 25 20:32:36 MDT 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jcomputchem2010.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Computational Chemistry",
  journal-URL =  "http://www.interscience.wiley.com/jpages/0192-8651",
  onlinedate =   "30 Jun 2015",
}


bibtype: book
filename: elefunt.bib
label: Temme:2015:AMI
author: Nico M. Temme
title: Asymptotic Methods for Integrals
volume: 6
publisher: pub-WORLD-SCI
address: pub-WORLD-SCI:adr
pages: xxii + 605
year: 2015
ISBN: 981-4612-15-4 (hardcover), 981-4612-16-2 (e-book) OR 9814612154 (hardcover), 9814612162 (ebook)
MRclass: 41-02 (33Cxx 33E20 65D30)
MRnumber: 3328507
MRreviewer: Jos{\'e} Luis L{\'o}pez
bibdate: Tue Feb 06 11:44:21 2018
bibtimestamp: 2018.02.06 11:44:21 ???
series: Series in Analysis
acknowledgement: ack-nhfb
rawauthor: Nico M. Temme
rawtitle: Asymptotic Methods for Integrals
ISBN13:
978-981-4612-15-9 (hardcover), 978-981-4612-16-6 (e-book) OR 9789814612159 (hardcover), 9789814612166 (ebook)

bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/numana2010.bib

abstract:
This book gives introductory chapters on the classical basic and standard methods for asymptotic analysis, such as Watson's lemma, Laplace's method, the saddle point and steepest descent methods, stationary phase and Darboux's method. The methods, explained in great detail, will obtain asymptotic approximations of the well-known special functions of mathematical physics and probability theory. After these introductory chapters, the methods of uniform asymptotic analysis are described in which several parameters have influence on typical phenomena: turning points and transition points, coinciding saddle and singularities. In all these examples, the special functions are indicated that describe the peculiar behavior of the integrals. The text extensively covers the classical methods with an emphasis on how to obtain expansions, and how to use the results for numerical methods, in particular for approximating special functions. In this way, we work with a computational mind: how can we use certain expansions in numerical analysis and in computer programs, how can we compute coefficients, and so on.

TOC:
Preface / vii Acknowledgments / ix Part 1: Basic Methods for Integrals / 1 1. Introduction / 3 1.1 Symbols used in asymptotic estimates / 3 1.2 Asymptotic expansions / 4 1.3 A first example: Exponential integral / 5 1.4 Generalized asymptotic expansions / 7 1.5 Properties of asymptotic power series / 8 1.6 Optimal truncation of asymptotic expansions / 10 2. Expansions of Laplace-type integrals: Watson's lemma / 13 2.1 Watson's lemma / 13 2.1.1 Watson's lemma for extended sectors / 14 2.1.2 More general forms of Watson's lemma / 16 2.2 Watson's lemma for loop integrals / 16 2.3 More general forms of Laplace-type integrals / 19 2.3.1 Transformation to the standard form / 19 2.4 How to compute the coefficients / 20 2.4.1 Inversion method for computing the coefficients / 20 2.4.2 Integrating by parts / 22 2.4.3 Manipulating power series / 23 2.4.4 Explicit forms of the coefficients in the expansion / 25 2.5 Other kernels / 26 2.6 Exponentially improved asymptotic expansions / 27 2.7 Singularities of the integrand / 29 2.7.1 A pole near the endpoint / 29 2.7.2 More general cases / 32 3. The method of Laplace / 33 3.1 A theorem for the general case / 33 3.2 Constructing the expansion / 35 3.2.1 Inversion method for computing the coefficients / 36 3.3 Explicit forms of the coefficients in the expansion / 37 3.4 The complementary error function / 38 4. The saddle point method and paths of steepest descent / 41 4.1 The axis of the valley at the saddle point / 43 4.2 Examples with simple exponentials / 43 4.2.1 A first example / 43 4.2.2 A cosine transform / 44 4.3 Steepest descent paths not through a saddle point / 44 4.3.1 A gamma function example / 45 4.3.2 An integral related to the error function / 46 4.4 An example with strong oscillations: A 100-digit challenge / 48 4.5 A Laplace inversion formula for $ erfc z $ / 49 4.6 A non-oscillatory integral for $ erfc z $, $ z in mathbb C $ / 50 4.7 The complex Airy function / 50 4.8 A parabolic cylinder function / 53 5. The Stokes phenomenon / 57 5.1 The Airy function / 57 5.2 The recent interest in the Stokes phenomenon / 58 5.3 Exponentially small terms in the Airy expansions / 59 5.4 Expansions in connection with the Stokes phenomenon / 60 5.4.1 Applications to a Kummer function / 61 Part 2: Basic Methods: Examples for Special Functions / 63 6. The gamma function / 65 6.1 $ Gamma (z) $ by Laplace's method / 66 6.1.1 Calculating the coefficients / 67 6.1.2 Details on the transformation / 68 6.2 $ 1 / Gamma (z) $ by the saddle point method / 71 6.2.1 Another integral representation of $ 1 / Gamma (z) $ / 72 6.3 The logarithm of the gamma function / 72 6.3.1 Estimations of the remainder / 73 6.4 Expansions of $ Gamma (z + a) $ and $ 1 / Gamma (z + a) $ / 75 6.5 The ratio of two gamma functions / 76 6.5.1 A simple expansion / 77 6.5.2 A more efficient expansion / 78 6.6 A binomial coefficient / 80 6.6.1 A uniform expansion of the binomial coefficient / 83 6.7 Asymptotic expansion of a product of gamma functions / 85 6.8 Expansions of ratios of three gamma functions / 88 7. Incomplete gamma functions / 91 7.1 Integral representations / 91 7.2 $ Gamma (a, x) $ : Asymptotic expansion for $ x gg a $ / 92 7.3 $ gamma (a, x) $ : Asymptotic expansion for $ a > x $ / 93 7.3.1 Singularity of the integrand / 94 7.3.2 More details on the transformation $ u = phi (t) $ / 96 7.4 $ Gamma (a, x) $ : Asymptotic expansion for $ x > a $ / 97 8. The Airy functions / 101 8.1 Expansions of $ Ai (z) $, $ Bi (z) $ / 102 8.1.1 Transforming the saddle point contour / 102 8.2 Expansions of $ Ai (-z) $, $ Bi (-z) $ / 105 8.3 Two simple ways to obtain the coefficients / 106 8.4 A generalized form of the Airy function / 107 9. Bessel functions: Large argument / 109 9.1 The modified Bessel function $ K_nu (z) $ / 109 9.2 The ordinary Bessel functions / 110 9.3 The modified Bessel function $ I_nu (z) $ / 111 9.3.1 A compound expansion of $ I_nu (z) $ / 111 9.4 Saddle point method for $ K_nu (z) $, $ z in mathbb C $ / 113 9.4.1 Integral representations from saddle point analysis / 115 9.4.2 Saddle point method for $ J_nu (x) $, $ x < nu $ / 116 9.5 Debye-type expansions of the modified Bessel functions / 117 9.6 Modified Bessel functions of purely imaginary order / 119 9.6.1 The monotonic case: $ x > nu > 0 $ / 120 9.6.2 The oscillatory case: $ nu > x > 0 $ / 123 9.7 A $ J $ -Bessel integral / 126 10. Kummer functions / 129 10.1 General properties / 129 10.2 Asymptotic expansions for large $ z $ / 131 10.3 Expansions for large $ a $ / 132 10.3.1 Tricomi's function $ E_nu (z) $ / 132 10.3.2 Expansion of $ U(a, c, z) $, $ a to +infty $ / 133 10.3.3 Expansion of $ _1F_1(a; c; z) $, $ a to +infty $ / 135 10.3.4 Expansion of $ _1F_1(a; c; z) $, $ a to -infty $ / 137 10.3.5 Expansion of $ U(a, c, z) $, $ a to -infty $ / 138 10.3.6 Slater's results for large $ a $ / 140 10.4 Expansions for large $ c $ / 142 10.4.1 Expansion of $ _1F_1(a; c; z) $, $ c to +infty $ / 142 10.4.2 Expansion of $ U(a, c, z) $, $ c to +infty $, $ z < c $ / 143 10.4.3 Expansion of $ U(a, c, z) $, $ c to +infty $, $ z > e $ / 144 10.4.4 Expansion of $ U(a, c, z) $, $ c to -infty $ / 145 10.4.5 Expansion of $ _1F_1(a; c; z) $, $ c to -infty $ / 147 10.5 Uniform expansions of the Kummer functions / 147 11. Parabolic cylinder functions: Large argument / 149 11.1 A few properties of the parabolic cylinder functions / 149 11.2 The function $ U(a, z) $ / 150 11.3 The function $ U(a, -z) $ / 152 11.4 The function $ V(a, z) $ / 153 11.5 Expansions of the derivatives / 154 12. The Gauss hypergeometric function / 155 12.1 Large values of $ c $ / 156 12.1.1 Large positive $ c $; $ |z| < z_0 $ / 156 12.1.2 Large negative $ c $; $ |z| < z_0 $ / 157 12.1.3 Large positive $ c $; $ |z| > z_0 $ / 158 12.1.4 Large negative $ c $; $ |z| > z_0 $ / 158 12.2 Large values of $ b $ / 158 12.2.1 Large negative $ b $; $ |z| > z_0 $ / 159 12.2.2 Large $ b $, $ |z| < z_0 $ / 159 12.3 Other large parameter cases / 160 12.3.1 Jacobi polynomials of large degree / 161 12.3.2 An example of the case $ _2F_1(a, b - lambda ; c + lambda ; z) $ / 163 13. Examples of $ _3F_2 $ -polynomials / 167 13.1 A $ _3F_2 $ associated with the Catalan--Larcombe--French sequence / 167 13.1.1 Transformations / 169 13.1.2 Asymptotic analysis / 170 13.1.3 Asymptotic expansion / 172 13.1.4 An alternative method / 173 13.2 An integral of Laguerre polynomials / 175 13.2.1 A first approach / 176 13.2.2 A generating function approach / 178 Part 3: Other Methods for Integrals / 181 14. The method of stationary phase / 183 14.1 Critical points / 183 14.2 Integrating by parts: No stationary points / 184 14.3 Three critical points: A formal approach / 185 14.4 On the use of neutralizes / 186 14.5 How to avoid neutralizes? / 188 14.5.1 A few details about the Fresnel integral / 190 14.6 Algebraic singularities at both endpoints: Erdelyi's example / 191 14.6.1 Application: A conical function / 192 14.6.2 Avoiding neutralizes in Erdelyi's example / 193 14.7 Transformation to standard form / 194 14.8 General order stationary points / 196 14.8.1 Integrating by parts / 196 14.9 The method fails: Examples / 197 14.9.1 The Airy function / 198 14.9.2 A more complicated example / 198 15. Coefficients of a power series. Darboux's method / 203 15.1 A first example: A binomial coefficient / 204 15.2 Legendre polynomials of large degree / 205 15.2.1 A paradox in asymptotics / 207 15.3 Gegenbauer polynomials of large degree / 208 15.4 Jacobi polynomials of large degree / 209 15.5 Laguerre polynomials of large degree / 209 15.6 Generalized Bernoulli polynomials $ B_n^(mu )(z) $ / 210 15.6.1 Asymptotic expansions for large degree / 211 15.6.2 An alternative expansion / 213 15.7 Generalized Euler polynomials $ E_n^(mu )(z) $ / 215 15.7.1 Asymptotic expansions for large degree / 215 15.7.2 An alternative expansion / 216 15.8 Coefficients of expansions of the $ _1F_1 $ -function / 218 15.8.1 Coefficients of Tricomi's expansion / 218 15.8.2 Coefficients of Buchholz's expansion / 221 16. Mellin--Barnes integrals and Mellin convolution integrals / 225 16.1 Mellin--Barnes integrals / 226 16.2 Mellin convolution integrals / 228 16.3 Error bounds / 230 17. Alternative expansions of Laplace-type integrals / 231 17.1 Hadamard-type expansions / 231 17.2 An expansion in terms of Kummer functions / 233 17.3 An expansion in terms of factorial series / 234 17.4 The Franklin--Friedman expansion / 237 18. Two-point Taylor expansions / 241 18.1 The expansions / 242 18.2 An alternative form of the expansion / 243 18.3 Explicit forms of the coefficients / 244 18.4 Manipulations with two-point Taylor expansions / 245 19. Hermite polynomials as limits of other classical orthogonal polynomials / 249 19.1 Limits between orthogonal polynomials / 249 19.2 The Askey scheme of orthogonal polynomials / 251 19.3 Asymptotic representations / 251 19.4 Gegenbauer polynomials / 253 19.5 Laguerre polynomials / 254 19.6 Generalized Bessel polynomials / 255 19.7 Meixner--Pollaczek polynomials into Laguerre polynomials / 257 Part 4: Uniform Methods for Integrals / 259 20. An overview of standard forms / 261 20.1 Comments on the table / 263 21. A saddle point near a pole / 267 21.1 A saddle point near a pole: Van der Waerden's method / 267 21.2 An alternative expansion / 269 21.3 An example from De Bruijn / 270 21.4 A pole near a double saddle point / 271 21.5 A singular perturbation problem and $ K $ -Bessel integrals / 272 21.5.1 A Bessel $ K_0 $ integral / 272 21.5.2 A similar Bessel $ K_1 $ integral / 274 21.5.3 A singular perturbation problem / 275 21.6 A double integral with poles near saddle points / 277 21.6.1 Application to a singular perturbation problem / 278 21.7 The Fermi--Dirac integral / 281 22. Saddle point near algebraic singularity / 285 22.1 A saddle point near an endpoint of the interval / 285 22.2 The Bleistein expansion / 286 22.3 Extending the role of the parameter /3 / 289 22.4 Contour integrals / 291 22.5 Kummer functions in terms of parabolic cylinder functions / 292 22.5.1 Uniform expansion of $ U(a, c, z) $, $ c to +infty $ / 293 22.5.2 Uniform expansion of $ _1F_1(a; c; z) $, $ c to +infty $ / 296 23. Two coalescing saddle points: Airy-type expansions / 299 23.1 The standard form / 299 23.2 An integration by parts method / 300 23.3 How to compute the coefficients / 302 23.4 An Airy-type expansion of the Hermite polynomial / 305 23.4.1 The cubic transformation / 306 23.4.2 Details on the coefficients / 308 23.5 An Airy-type expansion of the Bessel function $ J_nu (z) $ / 309 23.6 A semi-infinite interval: Incomplete Scorer function / 313 23.6.1 A singular perturbation problem inside a circle / 315 24. Hermite-type expansions of integrals / 319 24.1 An expansion in terms of Hermite polynomials / 320 24.1.1 Cauchy-type integrals for the coefficients / 321 24.2 Gegenbauer polynomials / 323 24.2.1 Preliminary steps / 324 24.2.2 A first approximation / 325 24.2.3 Transformation to the standard form / 326 24.2.4 Special cases of the expansion / 331 24.2.5 Approximating the zeros / 332 24.2.6 The relativistic Hermite polynomials / 333 24.3 Tricomi--Carlitz polynomials / 333 24.3.1 Contour integral and saddle points / 335 24.3.2 A first approximation / 337 24.3.3 Transformation to the standard form / 337 24.3.4 Approximating the zeros / 339 24.4 More examples / 340 Part 5: Uniform Methods for Laplace-Type Integrals / 341 25. The vanishing saddle point / 343 25.1 Expanding at the saddle point / 344 25.2 An integration by parts method / 346 25.2.1 Representing coefficients as a Cauchy-type integral / 347 25.3 Expansions for loop integrals / 348 25.4 Rummer functions / 350 25.5 Generalized zeta function / 350 25.6 Transforming to the standard form / 351 25.6.1 The ratio of two gamma functions / 352 25.6.2 Parabolic cylinder functions / 354 26. A moving endpoint: Incomplete Laplace integrals / 355 26.1 The standard form / 355 26.2 Constructing the expansion / 356 26.2.1 The complementary function / 357 26.3 Application to the incomplete beta function / 358 26.3.1 Expansions of the coefficients / 361 26.4 A corresponding loop integral / 362 26.4.1 Application to the incomplete beta function / 363 27. An essential singularity: Bessel-type expansions / 365 27.1 Expansions in terms of modified Bessel functions / 365 27.2 A corresponding loop integral / 368 27.3 Expansion at the internal saddle point / 368 27.4 Application to Kummer functions / 369 27.4.1 Expansion of $ U(a, c, z) $, $ a to +infty $, $ z > 0 $ / 369 27.4.2 Auxiliary expansions and further details / 372 27.4.3 Expansion of $ _1F_1(a: c; z) $, $ a to +infty $, $ z > 0 $ / 374 27.4.4 Expansion of $ _1F_1(a; c: z) $, $ a to -infty $, $ 0 < z < -4a $ / 375 27.4.5 Expansion of $ U(a, c, z) $, $ a to -infty $, $ 0 < z < -4a $ / 377 27.5 A second uniformity parameter / 378 27.5.1 Expansion of $ U(a, c, z) $, $ a to infty $, $ z > 0 $, $ c < 1 $ / 380 27.5.2 Expansion of $ _1F_1(a; c; z), $ a to infty $, $ z > 0 $, $ c > 1 $ / 381 28. Expansions in terms of Kummer functions / 383 28.1 Approximation in terms of the Kummer J7-function / 383 28.1.1 Constructing the expansions / 384 28.1.2 Expansion for the loop integral / 387 28.2 The $ _2F_1 $ function, large $ c $, in terms of $ U $ / 387 28.2.1 Legendre polynomials: Uniform expansions / 388 28.3 The $ _2F_1 $ -function, large $ b $ : in terms of $ _1F_1 $ / 389 28.3.1 Using a real integral / 390 28.3.2 Using a loop integral / 394 28.4 Jacobi polynomials of large degree: Laguerre-type expansion / 394 28.4.1 Laguerre-type expansion for large values of /3 / 398 28.5 Expansion of a Dirichlet-type integral / 401 Part 6: Uniform Examples for Special Functions / 403 29. Legendre functions / 405 29.1 Expansions of $ P_nu ^mu (z) $, $ Q_nu ^mu (z) $; $ nu to infty $, $ z geq 1 $ / 406 29.1.1 Expansions for $ z > z_0 > 1 $ / 400 29.1.2 Expansion in terms of modified Bessel functions / 407 29.1.3 Expansions of $ P_nu ^mu (z) $ and $ Q_nu ^mu (z) $ in terms of Bessel functions / 411 29.2 Expansions of $ P_nu ^mu (z) $, $ Q_nu ^mu (z) $; $ p to infty $, $ z > 1 $ / 412 29.2.1 Expansions for bounded $ z $ / 412 29.2.2 Expansions in terms of modified Bessel functions / 412 29.2.3 Expansions of $ P_nu ^mu (z) $ and $ Q_nu ^mu (z) $ / 413 29.3 Integrals with nearly coincident branch points / 414 29.3.1 Ursell's expansions of Legendre functions / 415 29.3.2 Coefficients of the expansion / 416 29.3.3 An alternative expansion of $ P_n^m(cosh z) $ / 417 29.3.4 A related integral with nearly coincident branch points / 418 29.4 Toroidal harmonics and conical functions / 418 30. Parabolic cylinder functions: Large parameter / 419 30.1 Notation for uniform asymptotic expansions / 419 30.2 The case $ a < 0 $ / 421 30.2.1 The case $ z > 2sqrt -a $ : $ -a + z to infty $ / 421 30.2.2 The case $ z < -2sqrt -a $ : $ -a - z to infty $ / 422 30.2.3 The case -2sqrt -a < z < 2sqrt -a / 423 30.3 The case $ a > 0 $ / 424 30.3.1 The case $ z > 0 $, $ a + z to infty $ / 425 30.3.2 The case $ z < 0 $, $ a - z to infty $ / 425 30.4 Expansions from integral representations / 426 30.4.1 The case $ a > 0 $, $ z > 0 $; $ a + z to infty $ / 426 30.4.2 The case $ a > 0 $, $ z < 0 $; $ a - z to infty $ / 428 30.4.3 The case $ a < 0 $, $ |z| > 2sqrt -a $; $ -a + |z| to infty $ / 429 30.5 Airy-type expansions / 430 31. Coulomb wave functions / 433 31.1 Contour integrals for Coulomb functions / 434 31.2 Expansions for $ rho to infty $ and bounded $ eta $ / / 435 31.3 Expansions for $ eta to infty $ and bounded $ rho $ / 437 31.4 Expansions for $ eta to -infty $ and bounded $ rho $ / 439 31.5 Expansions for $ eta to -infty and $ rho geq rho _0 > 0 $ / 440 31.6 Expansions for $ eta to -infty $ and $ rho geq 0 $ / 442 31.7 Expansions for $ eta $, $ rho to infty $; Airy-type expansions / 444 32. Laguerre polynomials: Uniform expansions / 449 32.1 An expansion for bounded $ z $ and $ a $ / 449 32.2 An expansion for bounded $ z $; $ a $ depends on $ n $ / 451 32.3 Expansions for bounded $ a $; $ z $ depends on $ n $ / 454 32.3.1 An expansion in terms of Airy functions / 455 32.3.2 An expansion in terms of Bessel functions / 456 32.4 An expansion in terms of Hermite polynomials; large $ a $ / 458 32.4.1 A first approximation / 459 32.4.2 Transformation to the standard form / 460 32.4.3 Approximating the zeros / 462 33. Generalized Bessel polynomials / 465 33.1 Relations to Bessel and Kummer functions / 466 33.2 An expansion in terms of Laguerre polynomials / 467 33.3 Expansions in terms of elementary functions / 470 33.3.1 The case $ |ph z| < pi /2 $ / 470 33.3.2 The case $ |ph (-z)| < pi /2 $ / 471 33.3.3 Integral representations / 472 33.3.4 Construction of the expansions / 472 33.4 Expansions in terms of modified Bessel functions / 476 33.4.1 Construction of the expansion / 476 34. Stirling numbers / 479 34.1 Definitions and integral representations / 479 34.2 Stirling number of the second kind / 481 34.2.1 Higher-order approximations / 483 34.2.2 About the positive saddle point / 486 34.2.3 About the quantity $ A $ / 487 34.3 Stirling numbers of the first kind / 488 35. Asymptotics of the integral $ int _0^1 cos (b x + a / x) dx $ / 491 35.1 The case $ b < a $ / 491 35.2 The case $ a = b $ / 493 35.3 The case $ b > a $ / 494 35.3.1 The contribution from $ mathcal P_1 $ / 495 35.3.2 The contribution from $ mathcal P_2 $ / 496 35.4 A Fresnel-type expansion / 497 Part 7: A Class of Cumulative Distribution Functions / 499 36. Expansions of a class of cumulative distribution functions / 501 36.1 Cumulative distribution functions: A standard form / 501 36.2 An incomplete normal distribution function / 505 36.3 The Sievert integral / 506 36.4 The Pearson type IV distribution / 507 36.5 The Von Mises distribution / 509 36.5.1 An expansion near the lower endpoint of integration / 511 37. Incomplete gamma functions: Uniform expansions / 513 37.1 Using the standard integral representations / 513 37.2 Representations by contour integrals / 514 37.2.1 Constructing the expansions / 516 37.2.2 Details on the coefficients / 518 37.2.3 Relations to the coefficients of earlier expansions / 520 37.3 Incomplete gamma functions, negative parameters / 520 37.3.1 Expansions near the transition point / 522 37.3.2 A real expansion of 7*(-a, -z) / 524 38. Incomplete beta function / 525 38.1 A power series expansion for large p / 526 38.2 A uniform expansion for large p / 526 38.3 The nearly symmetric case / 527 38.4 The general error function case / 529 39. Non-central chi-square, Marcum functions / 531 39.1 Properties of the Marcum functions / 532 39.2 More integral representations / 533 39.3 Asymptotic expansion; $ mu $ fixed, $ xi $ large / 535 39.4 Asymptotic expansion; $ xi + mu $ large / 537 39.5 An expansion in terms of the incomplete gamma function / 540 39.6 Comparison of the expansions numerically / 543 40. A weighted sum of exponentials / 545 40.1 An integral representation / 546 40.2 Saddle point analysis / 547 40.3 Details on the coefficients / 548 40.4 Auxiliary expansions / 550 40.5 Numerical verification / 551 41. A generalized incomplete gamma function / 553 41.1 An expansion in terms of incomplete gamma functions / 554 41.2 An expansion in terms of Laguerre polynomials / 554 41.3 An expansion in terms of Kummer functions / 555 41.4 An expansion in terms of the error function / 555 42. Asymptotic inversion of cumulative distribution functions / 559 42.1 The asymptotic inversion method / 559 42.2 Asymptotic inversion of the gamma distribution / 561 42.2.1 Numerical verification / 563 42.2.2 Other asymptotic inversion methods / 564 42.2.3 Asymptotics of the zeros of $ Gamma (a, z) $ / 565 42.3 Asymptotic inversion of the incomplete beta function / 567 42.3.1 Inverting by using the error function / 568 42.3.2 Inverting by using the incomplete gamma function / 569 42.3.3 Numerical verification / 572 42.4 The hyperbolic cumulative distribution / 573 42.4.1 Numerical verification / 574 42.5 The Marcum functions / 575 42.5.1 Asymptotic inversion / 576 42.5.2 Asymptotic inversion with respect to $ x $ / 576 42.5.3 Asymptotic inversion with respect to $ y $ / 579 Bibliography / 583 Index / 597

rawabstract:
This book gives introductory chapters on the classical basic and standard methods for asymptotic analysis, such as Watson's lemma, Laplace's method, the saddle point and steepest descent methods, stationary phase and Darboux's method. The methods, explained in great detail, will obtain asymptotic approximations of the well-known special functions of mathematical physics and probability theory. After these introductory chapters, the methods of uniform asymptotic analysis are described in which several parameters have influence on typical phenomena: turning points and transition points, coinciding saddle and singularities. In all these examples, the special functions are indicated that describe the peculiar behavior of the integrals. The text extensively covers the classical methods with an emphasis on how to obtain expansions, and how to use the results for numerical methods, in particular for approximating special functions. In this way, we work with a computational mind: how can we use certain expansions in numerical analysis and in computer programs, how can we compute coefficients, and so on.

shortTOC:
Basic methods for integrals Basic methods: examples for special functions Other methods for integrals Uniform methods for integrals Uniform methods for Laplace-type integrals Uniform examples for special functions A class of cumulative distribution factors

rawshortTOC:
Basic methods for integrals \\ Basic methods: examples for special functions \\ Other methods for integrals \\ Uniform methods for integrals \\ Uniform methods for Laplace-type integrals \\ Uniform examples for special functions \\ A class of cumulative distribution factors

rawTOC:
Preface / vii \\ Acknowledgments / ix \\ Part 1: Basic Methods for Integrals / 1 \\ 1. Introduction / 3 \\ 1.1 Symbols used in asymptotic estimates / 3 \\ 1.2 Asymptotic expansions / 4 \\ 1.3 A first example: Exponential integral / 5 \\ 1.4 Generalized asymptotic expansions / 7 \\ 1.5 Properties of asymptotic power series / 8 \\ 1.6 Optimal truncation of asymptotic expansions / 10 \\ 2. Expansions of Laplace-type integrals: Watson's lemma / 13 \\ 2.1 Watson's lemma / 13 \\ 2.1.1 Watson's lemma for extended sectors / 14 \\ 2.1.2 More general forms of Watson's lemma / 16 \\ 2.2 Watson's lemma for loop integrals / 16 \\ 2.3 More general forms of Laplace-type integrals / 19 \\ 2.3.1 Transformation to the standard form / 19 \\ 2.4 How to compute the coefficients / 20 \\ 2.4.1 Inversion method for computing the coefficients / 20 \\ 2.4.2 Integrating by parts / 22 \\ 2.4.3 Manipulating power series / 23 \\ 2.4.4 Explicit forms of the coefficients in the expansion / 25 \\ 2.5 Other kernels / 26 \\ 2.6 Exponentially improved asymptotic expansions / 27 \\ 2.7 Singularities of the integrand / 29 \\ 2.7.1 A pole near the endpoint / 29 \\ 2.7.2 More general cases / 32 \\ 3. The method of Laplace / 33 \\ 3.1 A theorem for the general case / 33 \\ 3.2 Constructing the expansion / 35 \\ 3.2.1 Inversion method for computing the coefficients / 36 \\ 3.3 Explicit forms of the coefficients in the expansion / 37 \\ 3.4 The complementary error function / 38 \\ 4. The saddle point method and paths of steepest descent / 41 \\ 4.1 The axis of the valley at the saddle point / 43 \\ 4.2 Examples with simple exponentials / 43 \\ 4.2.1 A first example / 43 \\ 4.2.2 A cosine transform / 44 \\ 4.3 Steepest descent paths not through a saddle point / 44 \\ 4.3.1 A gamma function example / 45 \\ 4.3.2 An integral related to the error function / 46 \\ 4.4 An example with strong oscillations: A 100-digit challenge / 48 \\ 4.5 A Laplace inversion formula for $ \erfc z $ / 49 \\ 4.6 A non-oscillatory integral for $ \erfc z $, $ z \in \mathbb{C} $ / 50 \\ 4.7 The complex Airy function / 50 \\ 4.8 A parabolic cylinder function / 53 \\ 5. The Stokes phenomenon / 57 \\ 5.1 The Airy function / 57 \\ 5.2 The recent interest in the Stokes phenomenon / 58 \\ 5.3 Exponentially small terms in the Airy expansions / 59 \\ 5.4 Expansions in connection with the Stokes phenomenon / 60 \\ 5.4.1 Applications to a Kummer function / 61 \\ Part 2: Basic Methods: Examples for Special Functions / 63 \\ 6. The gamma function / 65 \\ 6.1 $ \Gamma(z) $ by Laplace's method / 66 \\ 6.1.1 Calculating the coefficients / 67 \\ 6.1.2 Details on the transformation / 68 \\ 6.2 $ 1 / \Gamma(z) $ by the saddle point method / 71 \\ 6.2.1 Another integral representation of $ 1 / \Gamma(z) $ / 72 \\ 6.3 The logarithm of the gamma function / 72 \\ 6.3.1 Estimations of the remainder / 73 \\ 6.4 Expansions of $ \Gamma(z + a) $ and $ 1 / \Gamma(z + a) $ / 75 \\ 6.5 The ratio of two gamma functions / 76 \\ 6.5.1 A simple expansion / 77 \\ 6.5.2 A more efficient expansion / 78 \\ 6.6 A binomial coefficient / 80 \\ 6.6.1 A uniform expansion of the binomial coefficient / 83 \\ 6.7 Asymptotic expansion of a product of gamma functions / 85 \\ 6.8 Expansions of ratios of three gamma functions / 88 \\ 7. Incomplete gamma functions / 91 \\ 7.1 Integral representations / 91 \\ 7.2 $ \Gamma(a, x) $ : Asymptotic expansion for $ x \gg a $ / 92 \\ 7.3 $ \gamma(a, x) $ : Asymptotic expansion for $ a > x $ / 93 \\ 7.3.1 Singularity of the integrand / 94 \\ 7.3.2 More details on the transformation $ u = \phi(t) $ / 96 \\ 7.4 $ \Gamma(a, x) $ : Asymptotic expansion for $ x > a $ / 97 \\ 8. The Airy functions / 101 \\ 8.1 Expansions of $ \Ai(z) $, $ \Bi(z) $ / 102 \\ 8.1.1 Transforming the saddle point contour / 102 \\ 8.2 Expansions of $ \Ai(-z) $, $ \Bi(-z) $ / 105 \\ 8.3 Two simple ways to obtain the coefficients / 106 \\ 8.4 A generalized form of the Airy function / 107 \\ 9. Bessel functions: Large argument / 109 \\ 9.1 The modified Bessel function $ K_\nu(z) $ / 109 \\ 9.2 The ordinary Bessel functions / 110 \\ 9.3 The modified Bessel function $ I_\nu(z) $ / 111 9.3.1 A compound expansion of $ I_\nu(z) $ / 111 9.4 Saddle point method for $ K_\nu(z) $, $ z \in \mathbb{C} $ / 113 \\ 9.4.1 Integral representations from saddle point analysis / 115 \\ 9.4.2 Saddle point method for $ J_\nu(x) $, $ x < \nu $ / 116 \\ 9.5 Debye-type expansions of the modified Bessel functions / 117 \\ 9.6 Modified Bessel functions of purely imaginary order / 119 \\ 9.6.1 The monotonic case: $ x > \nu > 0 $ / 120 \\ 9.6.2 The oscillatory case: $ \nu > x > 0 $ / 123 \\ 9.7 A $ J $ -Bessel integral / 126 \\ 10. Kummer functions / 129 \\ 10.1 General properties / 129 \\ 10.2 Asymptotic expansions for large $ z $ / 131 \\ 10.3 Expansions for large $ a $ / 132 \\ 10.3.1 Tricomi's function $ E_\nu(z) $ / 132 \\ 10.3.2 Expansion of $ U(a, c, z) $, $ a \to +\infty $ / 133 \\ 10.3.3 Expansion of $ _1F_1(a; c; z) $, $ a \to +\infty $ / 135 \\ 10.3.4 Expansion of $ _1F_1(a; c; z) $, $ a \to -\infty $ / 137 \\ 10.3.5 Expansion of $ U(a, c, z) $, $ a \to -\infty $ / 138 \\ 10.3.6 Slater's results for large $ a $ / 140 \\ 10.4 Expansions for large $ c $ / 142 \\ 10.4.1 Expansion of $ _1F_1(a; c; z) $, $ c \to +\infty $ / 142 \\ 10.4.2 Expansion of $ U(a, c, z) $, $ c \to +\infty $, $ z < c $ / 143 \\ 10.4.3 Expansion of $ U(a, c, z) $, $ c \to +\infty $, $ z > e $ / 144 \\ 10.4.4 Expansion of $ U(a, c, z) $, $ c \to -\infty $ / 145 \\ 10.4.5 Expansion of $ _1F_1(a; c; z) $, $ c \to -\infty $ / 147 \\ 10.5 Uniform expansions of the Kummer functions / 147 \\ 11. Parabolic cylinder functions: Large argument / 149 \\ 11.1 A few properties of the parabolic cylinder functions / 149 \\ 11.2 The function $ U(a, z) $ / 150 \\ 11.3 The function $ U(a, -z) $ / 152 \\ 11.4 The function $ V(a, z) $ / 153 \\ 11.5 Expansions of the derivatives / 154 \\ 12. The Gauss hypergeometric function / 155 \\ 12.1 Large values of $ c $ / 156 \\ 12.1.1 Large positive $ c $; $ |z| < z_0 $ / 156 \\ 12.1.2 Large negative $ c $; $ |z| < z_0 $ / 157 \\ 12.1.3 Large positive $ c $; $ |z| > z_0 $ / 158 \\ 12.1.4 Large negative $ c $; $ |z| > z_0 $ / 158 \\ 12.2 Large values of $ b $ / 158 \\ 12.2.1 Large negative $ b $; $ |z| > z_0 $ / 159 \\ 12.2.2 Large $ b $, $ |z| < z_0 $ / 159 \\ 12.3 Other large parameter cases / 160 \\ 12.3.1 Jacobi polynomials of large degree / 161 \\ 12.3.2 An example of the case $ _2F_1(a, b - \lambda; c + \lambda; z) $ / 163 \\ 13. Examples of $ _3F_2 $ -polynomials / 167 \\ 13.1 A $ _3F_2 $ associated with the Catalan--Larcombe--French sequence / 167 \\ 13.1.1 Transformations / 169 \\ 13.1.2 Asymptotic analysis / 170 \\ 13.1.3 Asymptotic expansion / 172 \\ 13.1.4 An alternative method / 173 \\ 13.2 An integral of Laguerre polynomials / 175 \\ 13.2.1 A first approach / 176 \\ 13.2.2 A generating function approach / 178 \\ Part 3: Other Methods for Integrals / 181 \\ 14. The method of stationary phase / 183 \\ 14.1 Critical points / 183 \\ 14.2 Integrating by parts: No stationary points / 184 \\ 14.3 Three critical points: A formal approach / 185 \\ 14.4 On the use of neutralizes / 186 \\ 14.5 How to avoid neutralizes? / 188 \\ 14.5.1 A few details about the Fresnel integral / 190 \\ 14.6 Algebraic singularities at both endpoints: Erdelyi's example / 191 \\ 14.6.1 Application: A conical function / 192 \\ 14.6.2 Avoiding neutralizes in Erdelyi's example / 193 \\ 14.7 Transformation to standard form / 194 \\ 14.8 General order stationary points / 196 \\ 14.8.1 Integrating by parts / 196 \\ 14.9 The method fails: Examples / 197 \\ 14.9.1 The Airy function / 198 \\ 14.9.2 A more complicated example / 198 \\ 15. Coefficients of a power series. Darboux's method / 203 \\ 15.1 A first example: A binomial coefficient / 204 \\ 15.2 Legendre polynomials of large degree / 205 \\ 15.2.1 A paradox in asymptotics / 207 \\ 15.3 Gegenbauer polynomials of large degree / 208 \\ 15.4 Jacobi polynomials of large degree / 209 \\ 15.5 Laguerre polynomials of large degree / 209 \\ 15.6 Generalized Bernoulli polynomials $ B_n^{(\mu)}(z) $ / 210 \\ 15.6.1 Asymptotic expansions for large degree / 211 \\ 15.6.2 An alternative expansion / 213 \\ 15.7 Generalized Euler polynomials $ E_n^{(\mu)}(z) $ / 215 \\ 15.7.1 Asymptotic expansions for large degree / 215 \\ 15.7.2 An alternative expansion / 216 \\ 15.8 Coefficients of expansions of the $ _1F_1 $ -function / 218 \\ 15.8.1 Coefficients of Tricomi's expansion / 218 \\ 15.8.2 Coefficients of Buchholz's expansion / 221 \\ 16. Mellin--Barnes integrals and Mellin convolution integrals / 225 \\ 16.1 Mellin--Barnes integrals / 226 \\ 16.2 Mellin convolution integrals / 228 \\ 16.3 Error bounds / 230 \\ 17. Alternative expansions of Laplace-type integrals / 231 \\ 17.1 Hadamard-type expansions / 231 \\ 17.2 An expansion in terms of Kummer functions / 233 \\ 17.3 An expansion in terms of factorial series / 234 \\ 17.4 The Franklin--Friedman expansion / 237 \\ 18. Two-point Taylor expansions / 241 \\ 18.1 The expansions / 242 \\ 18.2 An alternative form of the expansion / 243 \\ 18.3 Explicit forms of the coefficients / 244 \\ 18.4 Manipulations with two-point Taylor expansions / 245 \\ 19. Hermite polynomials as limits of other classical orthogonal polynomials / 249 \\ 19.1 Limits between orthogonal polynomials / 249 \\ 19.2 The Askey scheme of orthogonal polynomials / 251 \\ 19.3 Asymptotic representations / 251 \\ 19.4 Gegenbauer polynomials / 253 \\ 19.5 Laguerre polynomials / 254 \\ 19.6 Generalized Bessel polynomials / 255 \\ 19.7 Meixner--Pollaczek polynomials into Laguerre polynomials / 257 \\ Part 4: Uniform Methods for Integrals / 259 \\ 20. An overview of standard forms / 261 \\ 20.1 Comments on the table / 263 \\ 21. A saddle point near a pole / 267 \\ 21.1 A saddle point near a pole: Van der Waerden's method / 267 \\ 21.2 An alternative expansion / 269 \\ 21.3 An example from De Bruijn / 270 \\ 21.4 A pole near a double saddle point / 271 \\ 21.5 A singular perturbation problem and $ K $ -Bessel integrals / 272 \\ 21.5.1 A Bessel $ K_0 $ integral / 272 \\ 21.5.2 A similar Bessel $ K_1 $ integral / 274 \\ 21.5.3 A singular perturbation problem / 275 \\ 21.6 A double integral with poles near saddle points / 277 \\ 21.6.1 Application to a singular perturbation problem / 278 \\ 21.7 The Fermi--Dirac integral / 281 \\ 22. Saddle point near algebraic singularity / 285 \\ 22.1 A saddle point near an endpoint of the interval / 285 \\ 22.2 The Bleistein expansion / 286 \\ 22.3 Extending the role of the parameter /3 / 289 \\ 22.4 Contour integrals / 291 \\ 22.5 Kummer functions in terms of parabolic cylinder functions / 292 \\ 22.5.1 Uniform expansion of $ U(a, c, z) $, $ c \to +\infty $ / 293 \\ 22.5.2 Uniform expansion of $ _1F_1(a; c; z) $, $ c \to +\infty $ / 296 \\ 23. Two coalescing saddle points: Airy-type expansions / 299 \\ 23.1 The standard form / 299 \\ 23.2 An integration by parts method / 300 \\ 23.3 How to compute the coefficients / 302 \\ 23.4 An Airy-type expansion of the Hermite polynomial / 305 \\ 23.4.1 The cubic transformation / 306 \\ 23.4.2 Details on the coefficients / 308 \\ 23.5 An Airy-type expansion of the Bessel function $ J_\nu(z) $ / 309 \\ 23.6 A semi-infinite interval: Incomplete Scorer function / 313 \\ 23.6.1 A singular perturbation problem inside a circle / 315 \\ 24. Hermite-type expansions of integrals / 319 \\ 24.1 An expansion in terms of Hermite polynomials / 320 \\ 24.1.1 Cauchy-type integrals for the coefficients / 321 \\ 24.2 Gegenbauer polynomials / 323 \\ 24.2.1 Preliminary steps / 324 \\ 24.2.2 A first approximation / 325 \\ 24.2.3 Transformation to the standard form / 326 \\ 24.2.4 Special cases of the expansion / 331 \\ 24.2.5 Approximating the zeros / 332 \\ 24.2.6 The relativistic Hermite polynomials / 333 \\ 24.3 Tricomi--Carlitz polynomials / 333 \\ 24.3.1 Contour integral and saddle points / 335 \\ 24.3.2 A first approximation / 337 \\ 24.3.3 Transformation to the standard form / 337 \\ 24.3.4 Approximating the zeros / 339 \\ 24.4 More examples / 340 \\ Part 5: Uniform Methods for Laplace-Type Integrals / 341 \\ 25. The vanishing saddle point / 343 \\ 25.1 Expanding at the saddle point / 344 \\ 25.2 An integration by parts method / 346 \\ 25.2.1 Representing coefficients as a Cauchy-type integral / 347 \\ 25.3 Expansions for loop integrals / 348 \\ 25.4 Rummer functions / 350 \\ 25.5 Generalized zeta function / 350 \\ 25.6 Transforming to the standard form / 351 \\ 25.6.1 The ratio of two gamma functions / 352 \\ 25.6.2 Parabolic cylinder functions / 354 \\ 26. A moving endpoint: Incomplete Laplace integrals / 355 \\ 26.1 The standard form / 355 \\ 26.2 Constructing the expansion / 356 \\ 26.2.1 The complementary function / 357 \\ 26.3 Application to the incomplete beta function / 358 \\ 26.3.1 Expansions of the coefficients / 361 \\ 26.4 A corresponding loop integral / 362 \\ 26.4.1 Application to the incomplete beta function / 363 \\ 27. An essential singularity: Bessel-type expansions / 365 \\ 27.1 Expansions in terms of modified Bessel functions / 365 \\ 27.2 A corresponding loop integral / 368 \\ 27.3 Expansion at the internal saddle point / 368 \\ 27.4 Application to Kummer functions / 369 \\ 27.4.1 Expansion of $ U(a, c, z) $, $ a \to +\infty $, $ z > 0 $ / 369 \\ 27.4.2 Auxiliary expansions and further details / 372 \\ 27.4.3 Expansion of $ _1F_1(a: c; z) $, $ a \to +\infty $, $ z > 0 $ / 374 \\ 27.4.4 Expansion of $ _1F_1(a; c: z) $, $ a \to -\infty $, $ 0 < z < -4a $ / 375 \\ 27.4.5 Expansion of $ U(a, c, z) $, $ a \to -\infty $, $ 0 < z < -4a $ / 377 \\ 27.5 A second uniformity parameter / 378 \\ 27.5.1 Expansion of $ U(a, c, z) $, $ a \to \infty $, $ z > 0 $, $ c < 1 $ / 380 \\ 27.5.2 Expansion of $ _1F_1(a; c; z), $ a \to \infty $, $ z > 0 $, $ c > 1 $ / 381 \\ 28. Expansions in terms of Kummer functions / 383 \\ 28.1 Approximation in terms of the Kummer J7-function / 383 \\ 28.1.1 Constructing the expansions / 384 \\ 28.1.2 Expansion for the loop integral / 387 \\ 28.2 The $ _2F_1 $ function, large $ c $, in terms of $ U $ / 387 \\ 28.2.1 Legendre polynomials: Uniform expansions / 388 \\ 28.3 The $ _2F_1 $ -function, large $ b $ : in terms of $ _1F_1 $ / 389 \\ 28.3.1 Using a real integral / 390 \\ 28.3.2 Using a loop integral / 394 \\ 28.4 Jacobi polynomials of large degree: Laguerre-type expansion / 394 \\ 28.4.1 Laguerre-type expansion for large values of /3 / 398 \\ 28.5 Expansion of a Dirichlet-type integral / 401 \\ Part 6: Uniform Examples for Special Functions / 403 \\ 29. Legendre functions / 405 \\ 29.1 Expansions of $ P_\nu^\mu(z) $, $ Q_\nu^\mu(z) $; $ \nu \to \infty $, $ z \geq 1 $ / 406 \\ 29.1.1 Expansions for $ z > z_0 > 1 $ / 400 \\ 29.1.2 Expansion in terms of modified Bessel functions / 407 \\ 29.1.3 Expansions of $ P_\nu^\mu(z) $ and $ Q_\nu^\mu(z) $ in terms of Bessel functions / 411 \\ 29.2 Expansions of $ P_\nu^\mu(z) $, $ Q_\nu^\mu(z) $; $ p \to \infty $, $ z > 1 $ / 412 \\ 29.2.1 Expansions for bounded $ z $ / 412 \\ 29.2.2 Expansions in terms of modified Bessel functions / 412 \\ 29.2.3 Expansions of $ P_\nu^\mu(z) $ and $ Q_\nu^\mu(z) $ / 413 \\ 29.3 Integrals with nearly coincident branch points / 414 \\ 29.3.1 Ursell's expansions of Legendre functions / 415 \\ 29.3.2 Coefficients of the expansion / 416 \\ 29.3.3 An alternative expansion of $ P_n^m(\cosh z) $ / 417 \\ 29.3.4 A related integral with nearly coincident branch points / 418 \\ 29.4 Toroidal harmonics and conical functions / 418 \\ 30. Parabolic cylinder functions: Large parameter / 419 \\ 30.1 Notation for uniform asymptotic expansions / 419 \\ 30.2 The case $ a < 0 $ / 421 \\ 30.2.1 The case $ z > 2\sqrt{-a} $ : $ -a + z \to \infty $ / 421 \\ 30.2.2 The case $ z < -2\sqrt{-a} $ : $ -a - z \to \infty $ / 422 \\ 30.2.3 The case -2\sqrt{-a} < z < 2\sqrt{-a} / 423 \\ 30.3 The case $ a > 0 $ / 424 \\ 30.3.1 The case $ z > 0 $, $ a + z \to \infty $ / 425 \\ 30.3.2 The case $ z < 0 $, $ a - z \to \infty $ / 425 \\ 30.4 Expansions from integral representations / 426 \\ 30.4.1 The case $ a > 0 $, $ z > 0 $; $ a + z \to \infty $ / 426 \\ 30.4.2 The case $ a > 0 $, $ z < 0 $; $ a - z \to \infty $ / 428 \\ 30.4.3 The case $ a < 0 $, $ |z| > 2\sqrt{-a} $; $ -a + |z| \to \infty $ / 429 \\ 30.5 Airy-type expansions / 430 \\ 31. Coulomb wave functions / 433 \\ 31.1 Contour integrals for Coulomb functions / 434 \\ 31.2 Expansions for $ \rho \to \infty $ and bounded $ \eta $ / / 435 \\ 31.3 Expansions for $ \eta \to \infty $ and bounded $ \rho $ / 437 \\ 31.4 Expansions for $ \eta \to -\infty $ and bounded $ \rho $ / 439 \\ 31.5 Expansions for $ \eta \to -\infty and $ \rho \geq \rho_0 > 0 $ / 440 \\ 31.6 Expansions for $ \eta \to -\infty $ and $ \rho \geq 0 $ / 442 \\ 31.7 Expansions for $ \eta $, $ \rho \to \infty $; Airy-type expansions / 444 \\ 32. Laguerre polynomials: Uniform expansions / 449 \\ 32.1 An expansion for bounded $ z $ and $ a $ / 449 \\ 32.2 An expansion for bounded $ z $; $ a $ depends on $ n $ / 451 \\ 32.3 Expansions for bounded $ a $; $ z $ depends on $ n $ / 454 \\ 32.3.1 An expansion in terms of Airy functions / 455 \\ 32.3.2 An expansion in terms of Bessel functions / 456 \\ 32.4 An expansion in terms of Hermite polynomials; large $ a $ / 458 \\ 32.4.1 A first approximation / 459 \\ 32.4.2 Transformation to the standard form / 460 \\ 32.4.3 Approximating the zeros / 462 \\ 33. Generalized Bessel polynomials / 465 \\ 33.1 Relations to Bessel and Kummer functions / 466 \\ 33.2 An expansion in terms of Laguerre polynomials / 467 \\ 33.3 Expansions in terms of elementary functions / 470 \\ 33.3.1 The case $ |\ph z| < \pi/2 $ / 470 \\ 33.3.2 The case $ |\ph(-z)| < \pi/2 $ / 471 \\ 33.3.3 Integral representations / 472 \\ 33.3.4 Construction of the expansions / 472 \\ 33.4 Expansions in terms of modified Bessel functions / 476 \\ 33.4.1 Construction of the expansion / 476 \\ 34. Stirling numbers / 479 \\ 34.1 Definitions and integral representations / 479 \\ 34.2 Stirling number of the second kind / 481 \\ 34.2.1 Higher-order approximations / 483 \\ 34.2.2 About the positive saddle point / 486 \\ 34.2.3 About the quantity $ A $ / 487 \\ 34.3 Stirling numbers of the first kind / 488 \\ 35. Asymptotics of the integral $ \int_0^1 \cos(b x + a / x) \, dx $ / 491 \\ 35.1 The case $ b < a $ / 491 \\ 35.2 The case $ a = b $ / 493 \\ 35.3 The case $ b > a $ / 494 \\ 35.3.1 The contribution from $ \mathcal{P}_1 $ / 495 \\ 35.3.2 The contribution from $ \mathcal{P}_2 $ / 496 \\ 35.4 A Fresnel-type expansion / 497 \\ Part 7: A Class of Cumulative Distribution Functions / 499 \\ 36. Expansions of a class of cumulative distribution functions / 501 \\ 36.1 Cumulative distribution functions: A standard form / 501 \\ 36.2 An incomplete normal distribution function / 505 \\ 36.3 The Sievert integral / 506 \\ 36.4 The Pearson type IV distribution / 507 \\ 36.5 The Von Mises distribution / 509 \\ 36.5.1 An expansion near the lower endpoint of integration / 511 \\ 37. Incomplete gamma functions: Uniform expansions / 513 \\ 37.1 Using the standard integral representations / 513 \\ 37.2 Representations by contour integrals / 514 \\ 37.2.1 Constructing the expansions / 516 \\ 37.2.2 Details on the coefficients / 518 \\ 37.2.3 Relations to the coefficients of earlier expansions / 520 \\ 37.3 Incomplete gamma functions, negative parameters / 520 \\ 37.3.1 Expansions near the transition point / 522 \\ 37.3.2 A real expansion of 7*(-a, -z) / 524 \\ 38. Incomplete beta function / 525 \\ 38.1 A power series expansion for large p / 526 \\ 38.2 A uniform expansion for large p / 526 \\ 38.3 The nearly symmetric case / 527 \\ 38.4 The general error function case / 529 \\ 39. Non-central chi-square, Marcum functions / 531 \\ 39.1 Properties of the Marcum functions / 532 \\ 39.2 More integral representations / 533 \\ 39.3 Asymptotic expansion; $ \mu $ fixed, $ \xi $ large / 535 \\ 39.4 Asymptotic expansion; $ \xi + \mu $ large / 537 \\ 39.5 An expansion in terms of the incomplete gamma function / 540 \\ 39.6 Comparison of the expansions numerically / 543 \\ 40. A weighted sum of exponentials / 545 \\ 40.1 An integral representation / 546 \\ 40.2 Saddle point analysis / 547 \\ 40.3 Details on the coefficients / 548 \\ 40.4 Auxiliary expansions / 550 \\ 40.5 Numerical verification / 551 \\ 41. A generalized incomplete gamma function / 553 \\ 41.1 An expansion in terms of incomplete gamma functions / 554 \\ 41.2 An expansion in terms of Laguerre polynomials / 554 \\ 41.3 An expansion in terms of Kummer functions / 555 \\ 41.4 An expansion in terms of the error function / 555 \\ 42. Asymptotic inversion of cumulative distribution functions / 559 \\ 42.1 The asymptotic inversion method / 559 \\ 42.2 Asymptotic inversion of the gamma distribution / 561 \\ 42.2.1 Numerical verification / 563 \\ 42.2.2 Other asymptotic inversion methods / 564 \\ 42.2.3 Asymptotics of the zeros of $ \Gamma(a, z) $ / 565 \\ 42.3 Asymptotic inversion of the incomplete beta function / 567 \\ 42.3.1 Inverting by using the error function / 568 \\ 42.3.2 Inverting by using the incomplete gamma function / 569 \\ 42.3.3 Numerical verification / 572 \\ 42.4 The hyperbolic cumulative distribution / 573 \\ 42.4.1 Numerical verification / 574 \\ 42.5 The Marcum functions / 575 \\ 42.5.1 Asymptotic inversion / 576 \\ 42.5.2 Asymptotic inversion with respect to $ x $ / 576 \\ 42.5.3 Asymptotic inversion with respect to $ y $ / 579 \\ Bibliography / 583 \\ Index / 597

entry:
@Book{Temme:2015:AMI,
  author =       "Nico M. Temme",
  title =        "Asymptotic Methods for Integrals",
  volume =       "6",
  publisher =    pub-WORLD-SCI,
  address =      pub-WORLD-SCI:adr,
  pages =        "xxii + 605",
  year =         "2015",
  ISBN =         "981-4612-15-4 (hardcover), 981-4612-16-2 (e-book)",
  ISBN-13 =      "978-981-4612-15-9 (hardcover), 978-981-4612-16-6
                 (e-book)",
  MRclass =      "41-02 (33Cxx 33E20 65D30)",
  MRnumber =     "3328507",
  MRreviewer =   "Jos{\'e} Luis L{\'o}pez",
  bibdate =      "Tue Feb 06 11:44:21 2018",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/numana2010.bib",
  series =       "Series in Analysis",
  abstract =     "This book gives introductory chapters on the classical
                 basic and standard methods for asymptotic analysis,
                 such as Watson's lemma, Laplace's method, the saddle
                 point and steepest descent methods, stationary phase
                 and Darboux's method. The methods, explained in great
                 detail, will obtain asymptotic approximations of the
                 well-known special functions of mathematical physics
                 and probability theory. After these introductory
                 chapters, the methods of uniform asymptotic analysis
                 are described in which several parameters have
                 influence on typical phenomena: turning points and
                 transition points, coinciding saddle and singularities.
                 In all these examples, the special functions are
                 indicated that describe the peculiar behavior of the
                 integrals. The text extensively covers the classical
                 methods with an emphasis on how to obtain expansions,
                 and how to use the results for numerical methods, in
                 particular for approximating special functions. In this
                 way, we work with a computational mind: how can we use
                 certain expansions in numerical analysis and in
                 computer programs, how can we compute coefficients, and
                 so on.",
  acknowledgement = ack-nhfb,
  shorttableofcontents = "Basic methods for integrals \\
                 Basic methods: examples for special functions \\
                 Other methods for integrals \\
                 Uniform methods for integrals \\
                 Uniform methods for Laplace-type integrals \\
                 Uniform examples for special functions \\
                 A class of cumulative distribution factors",
  tableofcontents = "Preface / vii \\
                 Acknowledgments / ix \\
                 Part 1: Basic Methods for Integrals / 1 \\
                 1. Introduction / 3 \\
                 1.1 Symbols used in asymptotic estimates / 3 \\
                 1.2 Asymptotic expansions / 4 \\
                 1.3 A first example: Exponential integral / 5 \\
                 1.4 Generalized asymptotic expansions / 7 \\
                 1.5 Properties of asymptotic power series / 8 \\
                 1.6 Optimal truncation of asymptotic expansions / 10
                 \\
                 2. Expansions of Laplace-type integrals: Watson's lemma
                 / 13 \\
                 2.1 Watson's lemma / 13 \\
                 2.1.1 Watson's lemma for extended sectors / 14 \\
                 2.1.2 More general forms of Watson's lemma / 16 \\
                 2.2 Watson's lemma for loop integrals / 16 \\
                 2.3 More general forms of Laplace-type integrals / 19
                 \\
                 2.3.1 Transformation to the standard form / 19 \\
                 2.4 How to compute the coefficients / 20 \\
                 2.4.1 Inversion method for computing the coefficients /
                 20 \\
                 2.4.2 Integrating by parts / 22 \\
                 2.4.3 Manipulating power series / 23 \\
                 2.4.4 Explicit forms of the coefficients in the
                 expansion / 25 \\
                 2.5 Other kernels / 26 \\
                 2.6 Exponentially improved asymptotic expansions / 27
                 \\
                 2.7 Singularities of the integrand / 29 \\
                 2.7.1 A pole near the endpoint / 29 \\
                 2.7.2 More general cases / 32 \\
                 3. The method of Laplace / 33 \\
                 3.1 A theorem for the general case / 33 \\
                 3.2 Constructing the expansion / 35 \\
                 3.2.1 Inversion method for computing the coefficients /
                 36 \\
                 3.3 Explicit forms of the coefficients in the expansion
                 / 37 \\
                 3.4 The complementary error function / 38 \\
                 4. The saddle point method and paths of steepest
                 descent / 41 \\
                 4.1 The axis of the valley at the saddle point / 43 \\
                 4.2 Examples with simple exponentials / 43 \\
                 4.2.1 A first example / 43 \\
                 4.2.2 A cosine transform / 44 \\
                 4.3 Steepest descent paths not through a saddle point /
                 44 \\
                 4.3.1 A gamma function example / 45 \\
                 4.3.2 An integral related to the error function / 46
                 \\
                 4.4 An example with strong oscillations: A 100-digit
                 challenge / 48 \\
                 4.5 A Laplace inversion formula for $ \erfc z $ / 49
                 \\
                 4.6 A non-oscillatory integral for $ \erfc z $, $ z \in
                 \mathbb{C} $ / 50 \\
                 4.7 The complex Airy function / 50 \\
                 4.8 A parabolic cylinder function / 53 \\
                 5. The Stokes phenomenon / 57 \\
                 5.1 The Airy function / 57 \\
                 5.2 The recent interest in the Stokes phenomenon / 58
                 \\
                 5.3 Exponentially small terms in the Airy expansions /
                 59 \\
                 5.4 Expansions in connection with the Stokes phenomenon
                 / 60 \\
                 5.4.1 Applications to a Kummer function / 61 \\
                 Part 2: Basic Methods: Examples for Special Functions /
                 63 \\
                 6. The gamma function / 65 \\
                 6.1 $ \Gamma(z) $ by Laplace's method / 66 \\
                 6.1.1 Calculating the coefficients / 67 \\
                 6.1.2 Details on the transformation / 68 \\
                 6.2 $ 1 / \Gamma(z) $ by the saddle point method / 71
                 \\
                 6.2.1 Another integral representation of $ 1 /
                 \Gamma(z) $ / 72 \\
                 6.3 The logarithm of the gamma function / 72 \\
                 6.3.1 Estimations of the remainder / 73 \\
                 6.4 Expansions of $ \Gamma(z + a) $ and $ 1 / \Gamma(z
                 + a) $ / 75 \\
                 6.5 The ratio of two gamma functions / 76 \\
                 6.5.1 A simple expansion / 77 \\
                 6.5.2 A more efficient expansion / 78 \\
                 6.6 A binomial coefficient / 80 \\
                 6.6.1 A uniform expansion of the binomial coefficient /
                 83 \\
                 6.7 Asymptotic expansion of a product of gamma
                 functions / 85 \\
                 6.8 Expansions of ratios of three gamma functions / 88
                 \\
                 7. Incomplete gamma functions / 91 \\
                 7.1 Integral representations / 91 \\
                 7.2 $ \Gamma(a, x) $ : Asymptotic expansion for $ x \gg
                 a $ / 92 \\
                 7.3 $ \gamma(a, x) $ : Asymptotic expansion for $ a > x
                 $ / 93 \\
                 7.3.1 Singularity of the integrand / 94 \\
                 7.3.2 More details on the transformation $ u = \phi(t)
                 $ / 96 \\
                 7.4 $ \Gamma(a, x) $ : Asymptotic expansion for $ x > a
                 $ / 97 \\
                 8. The Airy functions / 101 \\
                 8.1 Expansions of $ \Ai(z) $, $ \Bi(z) $ / 102 \\
                 8.1.1 Transforming the saddle point contour / 102 \\
                 8.2 Expansions of $ \Ai(-z) $, $ \Bi(-z) $ / 105 \\
                 8.3 Two simple ways to obtain the coefficients / 106
                 \\
                 8.4 A generalized form of the Airy function / 107 \\
                 9. Bessel functions: Large argument / 109 \\
                 9.1 The modified Bessel function $ K_\nu(z) $ / 109 \\
                 9.2 The ordinary Bessel functions / 110 \\
                 9.3 The modified Bessel function $ I_\nu(z) $ / 111
                 9.3.1 A compound expansion of $ I_\nu(z) $ / 111 9.4
                 Saddle point method for $ K_\nu(z) $, $ z \in
                 \mathbb{C} $ / 113 \\
                 9.4.1 Integral representations from saddle point
                 analysis / 115 \\
                 9.4.2 Saddle point method for $ J_\nu(x) $, $ x < \nu $
                 / 116 \\
                 9.5 Debye-type expansions of the modified Bessel
                 functions / 117 \\
                 9.6 Modified Bessel functions of purely imaginary order
                 / 119 \\
                 9.6.1 The monotonic case: $ x > \nu > 0 $ / 120 \\
                 9.6.2 The oscillatory case: $ \nu > x > 0 $ / 123 \\
                 9.7 A $ J $ -Bessel integral / 126 \\
                 10. Kummer functions / 129 \\
                 10.1 General properties / 129 \\
                 10.2 Asymptotic expansions for large $ z $ / 131 \\
                 10.3 Expansions for large $ a $ / 132 \\
                 10.3.1 Tricomi's function $ E_\nu(z) $ / 132 \\
                 10.3.2 Expansion of $ U(a, c, z) $, $ a \to +\infty $ /
                 133 \\
                 10.3.3 Expansion of $ _1F_1(a; c; z) $, $ a \to +\infty
                 $ / 135 \\
                 10.3.4 Expansion of $ _1F_1(a; c; z) $, $ a \to -\infty
                 $ / 137 \\
                 10.3.5 Expansion of $ U(a, c, z) $, $ a \to -\infty $ /
                 138 \\
                 10.3.6 Slater's results for large $ a $ / 140 \\
                 10.4 Expansions for large $ c $ / 142 \\
                 10.4.1 Expansion of $ _1F_1(a; c; z) $, $ c \to +\infty
                 $ / 142 \\
                 10.4.2 Expansion of $ U(a, c, z) $, $ c \to +\infty $,
                 $ z < c $ / 143 \\
                 10.4.3 Expansion of $ U(a, c, z) $, $ c \to +\infty $,
                 $ z > e $ / 144 \\
                 10.4.4 Expansion of $ U(a, c, z) $, $ c \to -\infty $ /
                 145 \\
                 10.4.5 Expansion of $ _1F_1(a; c; z) $, $ c \to -\infty
                 $ / 147 \\
                 10.5 Uniform expansions of the Kummer functions / 147
                 \\
                 11. Parabolic cylinder functions: Large argument / 149
                 \\
                 11.1 A few properties of the parabolic cylinder
                 functions / 149 \\
                 11.2 The function $ U(a, z) $ / 150 \\
                 11.3 The function $ U(a, -z) $ / 152 \\
                 11.4 The function $ V(a, z) $ / 153 \\
                 11.5 Expansions of the derivatives / 154 \\
                 12. The Gauss hypergeometric function / 155 \\
                 12.1 Large values of $ c $ / 156 \\
                 12.1.1 Large positive $ c $; $ |z| < z_0 $ / 156 \\
                 12.1.2 Large negative $ c $; $ |z| < z_0 $ / 157 \\
                 12.1.3 Large positive $ c $; $ |z| > z_0 $ / 158 \\
                 12.1.4 Large negative $ c $; $ |z| > z_0 $ / 158 \\
                 12.2 Large values of $ b $ / 158 \\
                 12.2.1 Large negative $ b $; $ |z| > z_0 $ / 159 \\
                 12.2.2 Large $ b $, $ |z| < z_0 $ / 159 \\
                 12.3 Other large parameter cases / 160 \\
                 12.3.1 Jacobi polynomials of large degree / 161 \\
                 12.3.2 An example of the case $ _2F_1(a, b - \lambda; c
                 + \lambda; z) $ / 163 \\
                 13. Examples of $ _3F_2 $ -polynomials / 167 \\
                 13.1 A $ _3F_2 $ associated with the
                 Catalan--Larcombe--French sequence / 167 \\
                 13.1.1 Transformations / 169 \\
                 13.1.2 Asymptotic analysis / 170 \\
                 13.1.3 Asymptotic expansion / 172 \\
                 13.1.4 An alternative method / 173 \\
                 13.2 An integral of Laguerre polynomials / 175 \\
                 13.2.1 A first approach / 176 \\
                 13.2.2 A generating function approach / 178 \\
                 Part 3: Other Methods for Integrals / 181 \\
                 14. The method of stationary phase / 183 \\
                 14.1 Critical points / 183 \\
                 14.2 Integrating by parts: No stationary points / 184
                 \\
                 14.3 Three critical points: A formal approach / 185 \\
                 14.4 On the use of neutralizes / 186 \\
                 14.5 How to avoid neutralizes? / 188 \\
                 14.5.1 A few details about the Fresnel integral / 190
                 \\
                 14.6 Algebraic singularities at both endpoints:
                 Erdelyi's example / 191 \\
                 14.6.1 Application: A conical function / 192 \\
                 14.6.2 Avoiding neutralizes in Erdelyi's example / 193
                 \\
                 14.7 Transformation to standard form / 194 \\
                 14.8 General order stationary points / 196 \\
                 14.8.1 Integrating by parts / 196 \\
                 14.9 The method fails: Examples / 197 \\
                 14.9.1 The Airy function / 198 \\
                 14.9.2 A more complicated example / 198 \\
                 15. Coefficients of a power series. Darboux's method /
                 203 \\
                 15.1 A first example: A binomial coefficient / 204 \\
                 15.2 Legendre polynomials of large degree / 205 \\
                 15.2.1 A paradox in asymptotics / 207 \\
                 15.3 Gegenbauer polynomials of large degree / 208 \\
                 15.4 Jacobi polynomials of large degree / 209 \\
                 15.5 Laguerre polynomials of large degree / 209 \\
                 15.6 Generalized Bernoulli polynomials $ B_n^{(\mu)}(z)
                 $ / 210 \\
                 15.6.1 Asymptotic expansions for large degree / 211 \\
                 15.6.2 An alternative expansion / 213 \\
                 15.7 Generalized Euler polynomials $ E_n^{(\mu)}(z) $ /
                 215 \\
                 15.7.1 Asymptotic expansions for large degree / 215 \\
                 15.7.2 An alternative expansion / 216 \\
                 15.8 Coefficients of expansions of the $ _1F_1 $
                 -function / 218 \\
                 15.8.1 Coefficients of Tricomi's expansion / 218 \\
                 15.8.2 Coefficients of Buchholz's expansion / 221 \\
                 16. Mellin--Barnes integrals and Mellin convolution
                 integrals / 225 \\
                 16.1 Mellin--Barnes integrals / 226 \\
                 16.2 Mellin convolution integrals / 228 \\
                 16.3 Error bounds / 230 \\
                 17. Alternative expansions of Laplace-type integrals /
                 231 \\
                 17.1 Hadamard-type expansions / 231 \\
                 17.2 An expansion in terms of Kummer functions / 233
                 \\
                 17.3 An expansion in terms of factorial series / 234
                 \\
                 17.4 The Franklin--Friedman expansion / 237 \\
                 18. Two-point Taylor expansions / 241 \\
                 18.1 The expansions / 242 \\
                 18.2 An alternative form of the expansion / 243 \\
                 18.3 Explicit forms of the coefficients / 244 \\
                 18.4 Manipulations with two-point Taylor expansions /
                 245 \\
                 19. Hermite polynomials as limits of other classical
                 orthogonal polynomials / 249 \\
                 19.1 Limits between orthogonal polynomials / 249 \\
                 19.2 The Askey scheme of orthogonal polynomials / 251
                 \\
                 19.3 Asymptotic representations / 251 \\
                 19.4 Gegenbauer polynomials / 253 \\
                 19.5 Laguerre polynomials / 254 \\
                 19.6 Generalized Bessel polynomials / 255 \\
                 19.7 Meixner--Pollaczek polynomials into Laguerre
                 polynomials / 257 \\
                 Part 4: Uniform Methods for Integrals / 259 \\
                 20. An overview of standard forms / 261 \\
                 20.1 Comments on the table / 263 \\
                 21. A saddle point near a pole / 267 \\
                 21.1 A saddle point near a pole: Van der Waerden's
                 method / 267 \\
                 21.2 An alternative expansion / 269 \\
                 21.3 An example from De Bruijn / 270 \\
                 21.4 A pole near a double saddle point / 271 \\
                 21.5 A singular perturbation problem and $ K $ -Bessel
                 integrals / 272 \\
                 21.5.1 A Bessel $ K_0 $ integral / 272 \\
                 21.5.2 A similar Bessel $ K_1 $ integral / 274 \\
                 21.5.3 A singular perturbation problem / 275 \\
                 21.6 A double integral with poles near saddle points /
                 277 \\
                 21.6.1 Application to a singular perturbation problem /
                 278 \\
                 21.7 The Fermi--Dirac integral / 281 \\
                 22. Saddle point near algebraic singularity / 285 \\
                 22.1 A saddle point near an endpoint of the interval /
                 285 \\
                 22.2 The Bleistein expansion / 286 \\
                 22.3 Extending the role of the parameter /3 / 289 \\
                 22.4 Contour integrals / 291 \\
                 22.5 Kummer functions in terms of parabolic cylinder
                 functions / 292 \\
                 22.5.1 Uniform expansion of $ U(a, c, z) $, $ c \to
                 +\infty $ / 293 \\
                 22.5.2 Uniform expansion of $ _1F_1(a; c; z) $, $ c \to
                 +\infty $ / 296 \\
                 23. Two coalescing saddle points: Airy-type expansions
                 / 299 \\
                 23.1 The standard form / 299 \\
                 23.2 An integration by parts method / 300 \\
                 23.3 How to compute the coefficients / 302 \\
                 23.4 An Airy-type expansion of the Hermite polynomial /
                 305 \\
                 23.4.1 The cubic transformation / 306 \\
                 23.4.2 Details on the coefficients / 308 \\
                 23.5 An Airy-type expansion of the Bessel function $
                 J_\nu(z) $ / 309 \\
                 23.6 A semi-infinite interval: Incomplete Scorer
                 function / 313 \\
                 23.6.1 A singular perturbation problem inside a circle
                 / 315 \\
                 24. Hermite-type expansions of integrals / 319 \\
                 24.1 An expansion in terms of Hermite polynomials / 320
                 \\
                 24.1.1 Cauchy-type integrals for the coefficients / 321
                 \\
                 24.2 Gegenbauer polynomials / 323 \\
                 24.2.1 Preliminary steps / 324 \\
                 24.2.2 A first approximation / 325 \\
                 24.2.3 Transformation to the standard form / 326 \\
                 24.2.4 Special cases of the expansion / 331 \\
                 24.2.5 Approximating the zeros / 332 \\
                 24.2.6 The relativistic Hermite polynomials / 333 \\
                 24.3 Tricomi--Carlitz polynomials / 333 \\
                 24.3.1 Contour integral and saddle points / 335 \\
                 24.3.2 A first approximation / 337 \\
                 24.3.3 Transformation to the standard form / 337 \\
                 24.3.4 Approximating the zeros / 339 \\
                 24.4 More examples / 340 \\
                 Part 5: Uniform Methods for Laplace-Type Integrals /
                 341 \\
                 25. The vanishing saddle point / 343 \\
                 25.1 Expanding at the saddle point / 344 \\
                 25.2 An integration by parts method / 346 \\
                 25.2.1 Representing coefficients as a Cauchy-type
                 integral / 347 \\
                 25.3 Expansions for loop integrals / 348 \\
                 25.4 Rummer functions / 350 \\
                 25.5 Generalized zeta function / 350 \\
                 25.6 Transforming to the standard form / 351 \\
                 25.6.1 The ratio of two gamma functions / 352 \\
                 25.6.2 Parabolic cylinder functions / 354 \\
                 26. A moving endpoint: Incomplete Laplace integrals /
                 355 \\
                 26.1 The standard form / 355 \\
                 26.2 Constructing the expansion / 356 \\
                 26.2.1 The complementary function / 357 \\
                 26.3 Application to the incomplete beta function / 358
                 \\
                 26.3.1 Expansions of the coefficients / 361 \\
                 26.4 A corresponding loop integral / 362 \\
                 26.4.1 Application to the incomplete beta function /
                 363 \\
                 27. An essential singularity: Bessel-type expansions /
                 365 \\
                 27.1 Expansions in terms of modified Bessel functions /
                 365 \\
                 27.2 A corresponding loop integral / 368 \\
                 27.3 Expansion at the internal saddle point / 368 \\
                 27.4 Application to Kummer functions / 369 \\
                 27.4.1 Expansion of $ U(a, c, z) $, $ a \to +\infty $,
                 $ z > 0 $ / 369 \\
                 27.4.2 Auxiliary expansions and further details / 372
                 \\
                 27.4.3 Expansion of $ _1F_1(a: c; z) $, $ a \to +\infty
                 $, $ z > 0 $ / 374 \\
                 27.4.4 Expansion of $ _1F_1(a; c: z) $, $ a \to -\infty
                 $, $ 0 < z < -4a $ / 375 \\
                 27.4.5 Expansion of $ U(a, c, z) $, $ a \to -\infty $,
                 $ 0 < z < -4a $ / 377 \\
                 27.5 A second uniformity parameter / 378 \\
                 27.5.1 Expansion of $ U(a, c, z) $, $ a \to \infty $, $
                 z > 0 $, $ c < 1 $ / 380 \\
                 27.5.2 Expansion of $ _1F_1(a; c; z), $ a \to \infty $,
                 $ z > 0 $, $ c > 1 $ / 381 \\
                 28. Expansions in terms of Kummer functions / 383 \\
                 28.1 Approximation in terms of the Kummer J7-function /
                 383 \\
                 28.1.1 Constructing the expansions / 384 \\
                 28.1.2 Expansion for the loop integral / 387 \\
                 28.2 The $ _2F_1 $ function, large $ c $, in terms of $
                 U $ / 387 \\
                 28.2.1 Legendre polynomials: Uniform expansions / 388
                 \\
                 28.3 The $ _2F_1 $ -function, large $ b $ : in terms of
                 $ _1F_1 $ / 389 \\
                 28.3.1 Using a real integral / 390 \\
                 28.3.2 Using a loop integral / 394 \\
                 28.4 Jacobi polynomials of large degree: Laguerre-type
                 expansion / 394 \\
                 28.4.1 Laguerre-type expansion for large values of /3 /
                 398 \\
                 28.5 Expansion of a Dirichlet-type integral / 401 \\
                 Part 6: Uniform Examples for Special Functions / 403
                 \\
                 29. Legendre functions / 405 \\
                 29.1 Expansions of $ P_\nu^\mu(z) $, $ Q_\nu^\mu(z) $;
                 $ \nu \to \infty $, $ z \geq 1 $ / 406 \\
                 29.1.1 Expansions for $ z > z_0 > 1 $ / 400 \\
                 29.1.2 Expansion in terms of modified Bessel functions
                 / 407 \\
                 29.1.3 Expansions of $ P_\nu^\mu(z) $ and $
                 Q_\nu^\mu(z) $ in terms of Bessel functions / 411 \\
                 29.2 Expansions of $ P_\nu^\mu(z) $, $ Q_\nu^\mu(z) $;
                 $ p \to \infty $, $ z > 1 $ / 412 \\
                 29.2.1 Expansions for bounded $ z $ / 412 \\
                 29.2.2 Expansions in terms of modified Bessel functions
                 / 412 \\
                 29.2.3 Expansions of $ P_\nu^\mu(z) $ and $
                 Q_\nu^\mu(z) $ / 413 \\
                 29.3 Integrals with nearly coincident branch points /
                 414 \\
                 29.3.1 Ursell's expansions of Legendre functions / 415
                 \\
                 29.3.2 Coefficients of the expansion / 416 \\
                 29.3.3 An alternative expansion of $ P_n^m(\cosh z) $ /
                 417 \\
                 29.3.4 A related integral with nearly coincident branch
                 points / 418 \\
                 29.4 Toroidal harmonics and conical functions / 418 \\
                 30. Parabolic cylinder functions: Large parameter / 419
                 \\
                 30.1 Notation for uniform asymptotic expansions / 419
                 \\
                 30.2 The case $ a < 0 $ / 421 \\
                 30.2.1 The case $ z > 2\sqrt{-a} $ : $ -a + z \to
                 \infty $ / 421 \\
                 30.2.2 The case $ z < -2\sqrt{-a} $ : $ -a - z \to
                 \infty $ / 422 \\
                 30.2.3 The case -2\sqrt{-a} < z < 2\sqrt{-a} / 423 \\
                 30.3 The case $ a > 0 $ / 424 \\
                 30.3.1 The case $ z > 0 $, $ a + z \to \infty $ / 425
                 \\
                 30.3.2 The case $ z < 0 $, $ a - z \to \infty $ / 425
                 \\
                 30.4 Expansions from integral representations / 426 \\
                 30.4.1 The case $ a > 0 $, $ z > 0 $; $ a + z \to
                 \infty $ / 426 \\
                 30.4.2 The case $ a > 0 $, $ z < 0 $; $ a - z \to
                 \infty $ / 428 \\
                 30.4.3 The case $ a < 0 $, $ |z| > 2\sqrt{-a} $; $ -a +
                 |z| \to \infty $ / 429 \\
                 30.5 Airy-type expansions / 430 \\
                 31. Coulomb wave functions / 433 \\
                 31.1 Contour integrals for Coulomb functions / 434 \\
                 31.2 Expansions for $ \rho \to \infty $ and bounded $
                 \eta $ / / 435 \\
                 31.3 Expansions for $ \eta \to \infty $ and bounded $
                 \rho $ / 437 \\
                 31.4 Expansions for $ \eta \to -\infty $ and bounded $
                 \rho $ / 439 \\
                 31.5 Expansions for $ \eta \to -\infty and $ \rho \geq
                 \rho_0 > 0 $ / 440 \\
                 31.6 Expansions for $ \eta \to -\infty $ and $ \rho
                 \geq 0 $ / 442 \\
                 31.7 Expansions for $ \eta $, $ \rho \to \infty $;
                 Airy-type expansions / 444 \\
                 32. Laguerre polynomials: Uniform expansions / 449 \\
                 32.1 An expansion for bounded $ z $ and $ a $ / 449 \\
                 32.2 An expansion for bounded $ z $; $ a $ depends on $
                 n $ / 451 \\
                 32.3 Expansions for bounded $ a $; $ z $ depends on $ n
                 $ / 454 \\
                 32.3.1 An expansion in terms of Airy functions / 455
                 \\
                 32.3.2 An expansion in terms of Bessel functions / 456
                 \\
                 32.4 An expansion in terms of Hermite polynomials;
                 large $ a $ / 458 \\
                 32.4.1 A first approximation / 459 \\
                 32.4.2 Transformation to the standard form / 460 \\
                 32.4.3 Approximating the zeros / 462 \\
                 33. Generalized Bessel polynomials / 465 \\
                 33.1 Relations to Bessel and Kummer functions / 466 \\
                 33.2 An expansion in terms of Laguerre polynomials /
                 467 \\
                 33.3 Expansions in terms of elementary functions / 470
                 \\
                 33.3.1 The case $ |\ph z| < \pi/2 $ / 470 \\
                 33.3.2 The case $ |\ph(-z)| < \pi/2 $ / 471 \\
                 33.3.3 Integral representations / 472 \\
                 33.3.4 Construction of the expansions / 472 \\
                 33.4 Expansions in terms of modified Bessel functions /
                 476 \\
                 33.4.1 Construction of the expansion / 476 \\
                 34. Stirling numbers / 479 \\
                 34.1 Definitions and integral representations / 479 \\
                 34.2 Stirling number of the second kind / 481 \\
                 34.2.1 Higher-order approximations / 483 \\
                 34.2.2 About the positive saddle point / 486 \\
                 34.2.3 About the quantity $ A $ / 487 \\
                 34.3 Stirling numbers of the first kind / 488 \\
                 35. Asymptotics of the integral $ \int_0^1 \cos(b x + a
                 / x) \, dx $ / 491 \\
                 35.1 The case $ b < a $ / 491 \\
                 35.2 The case $ a = b $ / 493 \\
                 35.3 The case $ b > a $ / 494 \\
                 35.3.1 The contribution from $ \mathcal{P}_1 $ / 495
                 \\
                 35.3.2 The contribution from $ \mathcal{P}_2 $ / 496
                 \\
                 35.4 A Fresnel-type expansion / 497 \\
                 Part 7: A Class of Cumulative Distribution Functions /
                 499 \\
                 36. Expansions of a class of cumulative distribution
                 functions / 501 \\
                 36.1 Cumulative distribution functions: A standard form
                 / 501 \\
                 36.2 An incomplete normal distribution function / 505
                 \\
                 36.3 The Sievert integral / 506 \\
                 36.4 The Pearson type IV distribution / 507 \\
                 36.5 The Von Mises distribution / 509 \\
                 36.5.1 An expansion near the lower endpoint of
                 integration / 511 \\
                 37. Incomplete gamma functions: Uniform expansions /
                 513 \\
                 37.1 Using the standard integral representations / 513
                 \\
                 37.2 Representations by contour integrals / 514 \\
                 37.2.1 Constructing the expansions / 516 \\
                 37.2.2 Details on the coefficients / 518 \\
                 37.2.3 Relations to the coefficients of earlier
                 expansions / 520 \\
                 37.3 Incomplete gamma functions, negative parameters /
                 520 \\
                 37.3.1 Expansions near the transition point / 522 \\
                 37.3.2 A real expansion of 7*(-a, -z) / 524 \\
                 38. Incomplete beta function / 525 \\
                 38.1 A power series expansion for large p / 526 \\
                 38.2 A uniform expansion for large p / 526 \\
                 38.3 The nearly symmetric case / 527 \\
                 38.4 The general error function case / 529 \\
                 39. Non-central chi-square, Marcum functions / 531 \\
                 39.1 Properties of the Marcum functions / 532 \\
                 39.2 More integral representations / 533 \\
                 39.3 Asymptotic expansion; $ \mu $ fixed, $ \xi $ large
                 / 535 \\
                 39.4 Asymptotic expansion; $ \xi + \mu $ large / 537
                 \\
                 39.5 An expansion in terms of the incomplete gamma
                 function / 540 \\
                 39.6 Comparison of the expansions numerically / 543 \\
                 40. A weighted sum of exponentials / 545 \\
                 40.1 An integral representation / 546 \\
                 40.2 Saddle point analysis / 547 \\
                 40.3 Details on the coefficients / 548 \\
                 40.4 Auxiliary expansions / 550 \\
                 40.5 Numerical verification / 551 \\
                 41. A generalized incomplete gamma function / 553 \\
                 41.1 An expansion in terms of incomplete gamma
                 functions / 554 \\
                 41.2 An expansion in terms of Laguerre polynomials /
                 554 \\
                 41.3 An expansion in terms of Kummer functions / 555
                 \\
                 41.4 An expansion in terms of the error function / 555
                 \\
                 42. Asymptotic inversion of cumulative distribution
                 functions / 559 \\
                 42.1 The asymptotic inversion method / 559 \\
                 42.2 Asymptotic inversion of the gamma distribution /
                 561 \\
                 42.2.1 Numerical verification / 563 \\
                 42.2.2 Other asymptotic inversion methods / 564 \\
                 42.2.3 Asymptotics of the zeros of $ \Gamma(a, z) $ /
                 565 \\
                 42.3 Asymptotic inversion of the incomplete beta
                 function / 567 \\
                 42.3.1 Inverting by using the error function / 568 \\
                 42.3.2 Inverting by using the incomplete gamma function
                 / 569 \\
                 42.3.3 Numerical verification / 572 \\
                 42.4 The hyperbolic cumulative distribution / 573 \\
                 42.4.1 Numerical verification / 574 \\
                 42.5 The Marcum functions / 575 \\
                 42.5.1 Asymptotic inversion / 576 \\
                 42.5.2 Asymptotic inversion with respect to $ x $ / 576
                 \\
                 42.5.3 Asymptotic inversion with respect to $ y $ / 579
                 \\
                 Bibliography / 583 \\
                 Index / 597",
}


bibtype: article
filename: elefunt.bib
label: Weiss:2015:ROS
author: Alexander K. H. Weiss and Christian Ochsenfeld
journal: j-J-COMPUT-CHEM
volume: 36
number: 18
pages: 1390--1398
day: 5
month: jul
monthnumber: 7
year: 2015
CODEN: JCCHDD
DOI: https://doi.org/10.1002/jcc.23935
ISSN: 0192-8651 (print), 1096-987X (electronic) OR 01928651 (print), 1096987X (electronic)
ISSNL: 0192-8651 OR 01928651
bibdate: Sat Jul 25 20:32:35 MDT 2015
bibtimestamp: 2015.07.25 20:32:35 MDT
fjournal: Journal of Computational Chemistry
acknowledgement: ack-nhfb
journalURL: http://www.interscience.wiley.com/jpages/0192-8651
onlinedate: 13 May 2015
rawauthor: Alexander K. H. Weiss and Christian Ochsenfeld
title:
A rigorous and optimized strategy for the evaluation of the Boys function kernel in molecular electronic structure theory

bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputchem2010.bib

rawtitle:
A rigorous and optimized strategy for the evaluation of the {Boys} function kernel in molecular electronic structure theory

entry:
@Article{Weiss:2015:ROS,
  author =       "Alexander K. H. Weiss and Christian Ochsenfeld",
  title =        "A rigorous and optimized strategy for the evaluation
                 of the {Boys} function kernel in molecular electronic
                 structure theory",
  journal =      j-J-COMPUT-CHEM,
  volume =       "36",
  number =       "18",
  pages =        "1390--1398",
  day =          "5",
  month =        jul,
  year =         "2015",
  CODEN =        "JCCHDD",
  DOI =          "https://doi.org/10.1002/jcc.23935",
  ISSN =         "0192-8651 (print), 1096-987X (electronic)",
  ISSN-L =       "0192-8651",
  bibdate =      "Sat Jul 25 20:32:35 MDT 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jcomputchem2010.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Computational Chemistry",
  journal-URL =  "http://www.interscience.wiley.com/jpages/0192-8651",
  onlinedate =   "13 May 2015",
}


bibtype: article
filename: elefunt.bib
label: Xu:2015:CFC
author: Ai-Min Xu and Zhong-Di Cen
title: Closed formulas for computing higher-order derivatives of functions involving exponential functions
journal: j-APPL-MATH-COMP
volume: 270
pages: 136--141
day: 1
month: nov
monthnumber: 11
year: 2015
CODEN: AMHCBQ
DOI: https://doi.org/10.1016/j.amc.2015.08.051
ISSN: 0096-3003 (print), 1873-5649 (electronic) OR 00963003 (print), 18735649 (electronic)
ISSNL: 0096-3003 OR 00963003
bibdate: Thu Nov 5 06:24:28 MST 2015
bibtimestamp: 2015.11.05 06:24:28 MST
URL: http://www.sciencedirect.com/science/article/pii/S0096300315011066
fjournal: Applied Mathematics and Computation
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00963003/
rawauthor: Ai-Min Xu and Zhong-Di Cen
rawtitle: Closed formulas for computing higher-order derivatives of functions involving exponential functions
keywords:
closed formula; derivatives of exponential functions; derivatives of trigonometric functions; higher-order derivative; hyperbolic function; tangent number; trigonometric function

remark:
The authors derive closed-form $n$-term sums for the $n$-th order derivatives of exponential and trigonometric functions. The sums involve factorials, powers, and Stirling numbers of the first and second kinds. At the end of their paper, they derive a new computationally-stable formula for the tangent numbers, $ T_2 n + 1 = sum _k = 1^n binom 2 n2 k - 1 T_2 k - 1 T_2(n - k) + 1$, a sum that involves only positive terms. There is a stable recurrence relation discussed in the MathCW book that is likely faster, because it requires only 2 multiplies and 1 add in each term of the recurrence.

entry:
@Article{Xu:2015:CFC,
  author =       "Ai-Min Xu and Zhong-Di Cen",
  title =        "Closed formulas for computing higher-order derivatives
                 of functions involving exponential functions",
  journal =      j-APPL-MATH-COMP,
  volume =       "270",
  number =       "??",
  pages =        "136--141",
  day =          "1",
  month =        nov,
  year =         "2015",
  CODEN =        "AMHCBQ",
  DOI =          "https://doi.org/10.1016/j.amc.2015.08.051",
  ISSN =         "0096-3003 (print), 1873-5649 (electronic)",
  ISSN-L =       "0096-3003",
  bibdate =      "Thu Nov 5 06:24:28 MST 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/applmathcomput2015.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0096300315011066",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied Mathematics and Computation",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00963003/",
  keywords =     "closed formula; derivatives of exponential functions;
                 derivatives of trigonometric functions; higher-order
                 derivative; hyperbolic function; tangent number;
                 trigonometric function",
  remark =       "The authors derive closed-form $n$-term sums for the
                 $n$-th order derivatives of exponential and
                 trigonometric functions. The sums involve factorials,
                 powers, and Stirling numbers of the first and second
                 kinds. At the end of their paper, they derive a new
                 computationally-stable formula for the tangent numbers,
                 $ T_{2 n + 1} = \sum_{k = 1}^n \binom {2 n}{2 k - 1}
                 T_{2 k - 1} T_{2(n - k) + 1}$, a sum that involves only
                 positive terms. There is a stable recurrence relation
                 discussed in the MathCW book that is likely faster,
                 because it requires only 2 multiplies and 1 add in each
                 term of the recurrence.",
}


bibtype: article
filename: elefunt.bib
label: Yang:2015:AFG
author: Zhen-Hang Yang and Yu-Ming Chu
title: Asymptotic formulas for gamma function with applications
journal: j-APPL-MATH-COMP
volume: 270
pages: 665--680
day: 1
month: nov
monthnumber: 11
year: 2015
CODEN: AMHCBQ
DOI: https://doi.org/10.1016/j.amc.2015.08.087
ISSN: 0096-3003 (print), 1873-5649 (electronic) OR 00963003 (print), 18735649 (electronic)
ISSNL: 0096-3003 OR 00963003
bibdate: Thu Nov 5 06:24:28 MST 2015
bibtimestamp: 2015.11.05 06:24:28 MST
URL: http://www.sciencedirect.com/science/article/pii/S0096300315011431
fjournal: Applied Mathematics and Computation
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00963003/
rawauthor: Zhen-Hang Yang and Yu-Ming Chu
rawtitle: Asymptotic formulas for gamma function with applications
entry:
@Article{Yang:2015:AFG,
  author =       "Zhen-Hang Yang and Yu-Ming Chu",
  title =        "Asymptotic formulas for gamma function with
                 applications",
  journal =      j-APPL-MATH-COMP,
  volume =       "270",
  number =       "??",
  pages =        "665--680",
  day =          "1",
  month =        nov,
  year =         "2015",
  CODEN =        "AMHCBQ",
  DOI =          "https://doi.org/10.1016/j.amc.2015.08.087",
  ISSN =         "0096-3003 (print), 1873-5649 (electronic)",
  ISSN-L =       "0096-3003",
  bibdate =      "Thu Nov 5 06:24:28 MST 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/applmathcomput2015.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0096300315011431",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied Mathematics and Computation",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00963003/",
}


bibtype: article
filename: elefunt.bib
label: Yang:2015:SBP
author: Zhen-Hang Yang and Yu-Ming Chu and Xiao-Hui Zhang
title: Sharp bounds for psi function
journal: j-APPL-MATH-COMP
volume: 268
pages: 1055--1063
day: 1
month: oct
monthnumber: 10
year: 2015
CODEN: AMHCBQ
DOI: https://doi.org/10.1016/j.amc.2015.07.012
ISSN: 0096-3003 (print), 1873-5649 (electronic) OR 00963003 (print), 18735649 (electronic)
ISSNL: 0096-3003 OR 00963003
bibdate: Wed Sep 16 06:56:32 MDT 2015
bibtimestamp: 2015.09.16 06:56:32 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0096300315009248
fjournal: Applied Mathematics and Computation
keywords: Gamma function; Monotonicity; Psi function
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00963003/
rawauthor: Zhen-Hang Yang and Yu-Ming Chu and Xiao-Hui Zhang
rawtitle: Sharp bounds for psi function
entry:
@Article{Yang:2015:SBP,
  author =       "Zhen-Hang Yang and Yu-Ming Chu and Xiao-Hui Zhang",
  title =        "Sharp bounds for psi function",
  journal =      j-APPL-MATH-COMP,
  volume =       "268",
  number =       "??",
  pages =        "1055--1063",
  day =          "1",
  month =        oct,
  year =         "2015",
  CODEN =        "AMHCBQ",
  DOI =          "https://doi.org/10.1016/j.amc.2015.07.012",
  ISSN =         "0096-3003 (print), 1873-5649 (electronic)",
  ISSN-L =       "0096-3003",
  bibdate =      "Wed Sep 16 06:56:32 MDT 2015",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/applmathcomput2015.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0096300315009248",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied Mathematics and Computation",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00963003/",
  keywords =     "Gamma function; Monotonicity; Psi function",
}


bibtype: article
filename: elefunt.bib
label: Zhang:2015:EAR
author: Jianfeng Zhang and Paul Chow and Hengzhu Liu
title: An Enhanced Adaptive Recoding Rotation CORDIC
journal: j-TRETS
volume: 9
number: 1
pages: 4:1--4:??
month: nov
monthnumber: 11
year: 2015
DOI: https://doi.org/10.1145/2812813
ISSN: 1936-7406 (print), 1936-7414 (electronic) OR 19367406 (print), 19367414 (electronic)
ISSNL: 1936-7406 OR 19367406
bibdate: Tue Dec 22 16:19:56 MST 2015
bibtimestamp: 2015.12.22 16:19:56 MST
fjournal: ACM Transactions on Reconfigurable Technology and Systems (TRETS)
acknowledgement: ack-nhfb
articleno: 4
journalURL: http://portal.acm.org/toc.cfm?id=J1151
rawauthor: Jianfeng Zhang and Paul Chow and Hengzhu Liu
rawtitle: An Enhanced Adaptive Recoding Rotation {CORDIC}
bibsource:
http://portal.acm.org/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/trets.bib

abstract:
The Conventional Coordinate Rotation Digital Computer (CORDIC) algorithm has been widely used in many applications, particularly in Direct Digital Frequency Synthesizers (DDS) and Fast Fourier Transforms (FFT). However, CORDIC is constrained by the excessive number of iterations, angle data path, and scaling factor compensation. In this article, an enhanced adaptive recoding CORDIC (EARC) is proposed. It uses the enhanced adaptive recoding method to reduce the required iterations and adopts the trigonometric transformation scheme to scale up the rotation angles. Computing sine and cosine is used first to compare the core functionality of EARC with basic CORDIC; then a 16-bit DDS and a 1,024-point FFT based on EARC are evaluated to demonstrate the benefits of EARC in larger applications. All the proposed architectures are validated on a Virtex 5 FPGA development platform. Compared with a commercial implementation of CORDIC, EARC requires 33.3% less hardware resources, provides a twofold speedup, dissipates 70.4% less power, and improves accuracy in terms of the Bit Error Position (BEP). Compared to the state-of-the-art Hybrid CORDIC, EARC reduces latency by 11.1% and consumes 17% less power. Compared with a commercial implementation of DDS, the dissipated power of the proposed DDS is reduced by 27.2%. The proposed DDS improves Spurious-Free Dynamic Range (SFDR) by nearly 7 dBc and dissipates 21.8% less power when compared with a recently published DDS circuit. The FFT based on EARC dissipates a factor of 2.05 less power than the commercial FFT even when choosing the 100% toggle rate for the FFT based on EARC and the 12.5% toggle rate for the commercial FFT. Compared with a recently published FFT, the FFT based on EARC improves Signal-to-Noise Ratio (SNR) by 8.9 dB and consumes 7.78% less power.

rawabstract:
The Conventional Coordinate Rotation Digital Computer (CORDIC) algorithm has been widely used in many applications, particularly in Direct Digital Frequency Synthesizers (DDS) and Fast Fourier Transforms (FFT). However, CORDIC is constrained by the excessive number of iterations, angle data path, and scaling factor compensation. In this article, an enhanced adaptive recoding CORDIC (EARC) is proposed. It uses the enhanced adaptive recoding method to reduce the required iterations and adopts the trigonometric transformation scheme to scale up the rotation angles. Computing sine and cosine is used first to compare the core functionality of EARC with basic CORDIC; then a 16-bit DDS and a 1,024-point FFT based on EARC are evaluated to demonstrate the benefits of EARC in larger applications. All the proposed architectures are validated on a Virtex 5 FPGA development platform. Compared with a commercial implementation of CORDIC, EARC requires 33.3\% less hardware resources, provides a twofold speedup, dissipates 70.4\% less power, and improves accuracy in terms of the Bit Error Position (BEP). Compared to the state-of-the-art Hybrid CORDIC, EARC reduces latency by 11.1\% and consumes 17\% less power. Compared with a commercial implementation of DDS, the dissipated power of the proposed DDS is reduced by 27.2\%. The proposed DDS improves Spurious-Free Dynamic Range (SFDR) by nearly 7 dBc and dissipates 21.8\% less power when compared with a recently published DDS circuit. The FFT based on EARC dissipates a factor of 2.05 less power than the commercial FFT even when choosing the 100\% toggle rate for the FFT based on EARC and the 12.5\% toggle rate for the commercial FFT. Compared with a recently published FFT, the FFT based on EARC improves Signal-to-Noise Ratio (SNR) by 8.9 dB and consumes 7.78\% less power.

entry:
@Article{Zhang:2015:EAR,
  author =       "Jianfeng Zhang and Paul Chow and Hengzhu Liu",
  title =        "An Enhanced Adaptive Recoding Rotation {CORDIC}",
  journal =      j-TRETS,
  volume =       "9",
  number =       "1",
  pages =        "4:1--4:??",
  month =        nov,
  year =         "2015",
  CODEN =        "????",
  DOI =          "https://doi.org/10.1145/2812813",
  ISSN =         "1936-7406 (print), 1936-7414 (electronic)",
  ISSN-L =       "1936-7406",
  bibdate =      "Tue Dec 22 16:19:56 MST 2015",
  bibsource =    "http://portal.acm.org/;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/trets.bib",
  abstract =     "The Conventional Coordinate Rotation Digital Computer
                 (CORDIC) algorithm has been widely used in many
                 applications, particularly in Direct Digital Frequency
                 Synthesizers (DDS) and Fast Fourier Transforms (FFT).
                 However, CORDIC is constrained by the excessive number
                 of iterations, angle data path, and scaling factor
                 compensation. In this article, an enhanced adaptive
                 recoding CORDIC (EARC) is proposed. It uses the
                 enhanced adaptive recoding method to reduce the
                 required iterations and adopts the trigonometric
                 transformation scheme to scale up the rotation angles.
                 Computing sine and cosine is used first to compare the
                 core functionality of EARC with basic CORDIC; then a
                 16-bit DDS and a 1,024-point FFT based on EARC are
                 evaluated to demonstrate the benefits of EARC in larger
                 applications. All the proposed architectures are
                 validated on a Virtex 5 FPGA development platform.
                 Compared with a commercial implementation of CORDIC,
                 EARC requires 33.3\% less hardware resources, provides
                 a twofold speedup, dissipates 70.4\% less power, and
                 improves accuracy in terms of the Bit Error Position
                 (BEP). Compared to the state-of-the-art Hybrid CORDIC,
                 EARC reduces latency by 11.1\% and consumes 17\% less
                 power. Compared with a commercial implementation of
                 DDS, the dissipated power of the proposed DDS is
                 reduced by 27.2\%. The proposed DDS improves
                 Spurious-Free Dynamic Range (SFDR) by nearly 7 dBc and
                 dissipates 21.8\% less power when compared with a
                 recently published DDS circuit. The FFT based on EARC
                 dissipates a factor of 2.05 less power than the
                 commercial FFT even when choosing the 100\% toggle rate
                 for the FFT based on EARC and the 12.5\% toggle rate
                 for the commercial FFT. Compared with a recently
                 published FFT, the FFT based on EARC improves
                 Signal-to-Noise Ratio (SNR) by 8.9 dB and consumes
                 7.78\% less power.",
  acknowledgement = ack-nhfb,
  articleno =    "4",
  fjournal =     "ACM Transactions on Reconfigurable Technology and
                 Systems (TRETS)",
  journal-URL =  "http://portal.acm.org/toc.cfm?id=J1151",
}


bibtype: article
filename: elefunt.bib
label: Abel:2016:HOA
author: Ulrich Abel
title: High order algorithms for calculating roots
journal: j-MATH-GAZ
volume: 100
number: 549
pages: 420--428
month: nov
monthnumber: 11
year: 2016
CODEN: MAGAAS
DOI: https://doi.org/10.1017/mag.2016.106
ISSN: 0025-5572 (print), 2056-6328 (electronic) OR 00255572 (print), 20566328 (electronic)
ISSNL: 0025-5572 OR 00255572
bibdate: Thu Nov 17 10:32:54 MST 2016
bibtimestamp: 2016.11.17 10:32:54 MST
URL: https://www.cambridge.org/core/product/2FACD442DB78364B04DA2E64BA06F269
fjournal: The Mathematical Gazette
acknowledgement: ack-nhfb
ajournal: Math. Gaz.
onlinedate: 17 October 2016
rawauthor: Ulrich Abel
rawtitle: High order algorithms for calculating roots
bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathgaz2010.bib

journalURL:
http://journals.cambridge.org/action/displayIssue?jid=MAG; http://www.m-a.org.uk/jsp/index.jsp?lnk=620

entry:
@Article{Abel:2016:HOA,
  author =       "Ulrich Abel",
  title =        "High order algorithms for calculating roots",
  journal =      j-MATH-GAZ,
  volume =       "100",
  number =       "549",
  pages =        "420--428",
  month =        nov,
  year =         "2016",
  CODEN =        "MAGAAS",
  DOI =          "https://doi.org/10.1017/mag.2016.106",
  ISSN =         "0025-5572 (print), 2056-6328 (electronic)",
  ISSN-L =       "0025-5572",
  bibdate =      "Thu Nov 17 10:32:54 MST 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/mathgaz2010.bib",
  URL =          "https://www.cambridge.org/core/product/2FACD442DB78364B04DA2E64BA06F269",
  acknowledgement = ack-nhfb,
  ajournal =     "Math. Gaz.",
  fjournal =     "The Mathematical Gazette",
  journal-URL =  "http://journals.cambridge.org/action/displayIssue?jid=MAG;
                 http://www.m-a.org.uk/jsp/index.jsp?lnk=620",
  onlinedate =   "17 October 2016",
}


bibtype: article
filename: elefunt.bib
label: Aprahamian:2016:MIT
author: Mary Aprahamian and Nicholas J. Higham
title: Matrix Inverse Trigonometric and Inverse Hyperbolic Functions: Theory and Algorithms
journal: j-SIAM-J-MAT-ANA-APPL
volume: 37
number: 4
pages: 1453--1477
year: 2016
CODEN: SJMAEL
DOI: https://doi.org/10.1137/16M1057577
ISSN: 0895-4798 (print), 1095-7162 (electronic) OR 08954798 (print), 10957162 (electronic)
ISSNL: 0895-4798 OR 08954798
bibdate: Fri Aug 25 09:01:43 MDT 2017
bibtimestamp: 2017.08.25 09:01:43 MDT
fjournal: SIAM Journal on Matrix Analysis and Applications
acknowledgement: ack-nhfb
journalURL: http://epubs.siam.org/simax
onlinedate: January 2016
rawauthor: Mary Aprahamian and Nicholas J. Higham
rawtitle: Matrix Inverse Trigonometric and Inverse Hyperbolic Functions: Theory and Algorithms
bibsource:
http://epubs.siam.org/sam-bin/dbq/toc/SIMAX/37/4; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/siamjmatanaappl.bib

entry:
@Article{Aprahamian:2016:MIT,
  author =       "Mary Aprahamian and Nicholas J. Higham",
  title =        "Matrix Inverse Trigonometric and Inverse Hyperbolic
                 Functions: Theory and Algorithms",
  journal =      j-SIAM-J-MAT-ANA-APPL,
  volume =       "37",
  number =       "4",
  pages =        "1453--1477",
  month =        "????",
  year =         "2016",
  CODEN =        "SJMAEL",
  DOI =          "https://doi.org/10.1137/16M1057577",
  ISSN =         "0895-4798 (print), 1095-7162 (electronic)",
  ISSN-L =       "0895-4798",
  bibdate =      "Fri Aug 25 09:01:43 MDT 2017",
  bibsource =    "http://epubs.siam.org/sam-bin/dbq/toc/SIMAX/37/4;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/siamjmatanaappl.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "SIAM Journal on Matrix Analysis and Applications",
  journal-URL =  "http://epubs.siam.org/simax",
  onlinedate =   "January 2016",
}


bibtype: article
filename: elefunt.bib
label: Bailey:2016:CSC
author: D. H. Bailey and J. M. Borwein
journal: j-MATH-COMPUT
volume: 85
number: 297
pages: 295--324
year: 2016
CODEN: MCMPAF
DOI: https://doi.org/10.1090/mcom/2974
ISSN: 0025-5718 (print), 1088-6842 (electronic) OR 00255718 (print), 10886842 (electronic)
ISSNL: 0025-5718 OR 00255718
bibdate: Mon Feb 8 17:02:07 MST 2016
bibtimestamp: 2016.02.08 17:02:07 MST
fjournal: Mathematics of Computation
acknowledgement: ack-nhfb
journalURL: http://www.ams.org/mcom/
rawauthor: D. H. Bailey and J. M. Borwein
title:
Computation and structure of character polylogarithms with applications to character Mordell--Tornheim--Witten sums

bibsource:
http://www.ams.org/mcom/2016-85-297; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp2010.bib

URL:
http://www.ams.org/journals/mcom/2016-85-297/S0025-5718-2015-02974-3; http://www.ams.org/journals/mcom/2016-85-297/S0025-5718-2015-02974-3/S0025-5718-2015-02974-3.pdf; http://www.ams.org/mathscinet/search/author.html?authorName=Borwein%2C%20J.%20M; http://www.ams.org/mathscinet/search/author.html?mrauthid=29355

rawtitle:
Computation and structure of character polylogarithms with applications to character {Mordell--Tornheim--Witten} sums

entry:
@Article{Bailey:2016:CSC,
  author =       "D. H. Bailey and J. M. Borwein",
  title =        "Computation and structure of character polylogarithms
                 with applications to character
                 {Mordell--Tornheim--Witten} sums",
  journal =      j-MATH-COMPUT,
  volume =       "85",
  number =       "297",
  pages =        "295--324",
  month =        "",
  year =         "2016",
  CODEN =        "MCMPAF",
  DOI =          "https://doi.org/10.1090/mcom/2974",
  ISSN =         "0025-5718 (print), 1088-6842 (electronic)",
  ISSN-L =       "0025-5718",
  bibdate =      "Mon Feb 8 17:02:07 MST 2016",
  bibsource =    "http://www.ams.org/mcom/2016-85-297;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/mathcomp2010.bib",
  URL =          "http://www.ams.org/journals/mcom/2016-85-297/S0025-5718-2015-02974-3;
                 http://www.ams.org/journals/mcom/2016-85-297/S0025-5718-2015-02974-3/S0025-5718-2015-02974-3.pdf;
                 http://www.ams.org/mathscinet/search/author.html?authorName=Borwein%2C%20J.%20M;
                 http://www.ams.org/mathscinet/search/author.html?mrauthid=29355",
  acknowledgement = ack-nhfb,
  fjournal =     "Mathematics of Computation",
  journal-URL =  "http://www.ams.org/mcom/",
}


bibtype: article
filename: elefunt.bib
label: Bao:2016:SAO
author: Vo Nguyen Quoc Bao and Luu Pham Tuyen and Huynh Huu Tue
title: A Survey on Approximations of One-Dimensional Gaussian $Q$-Function
journal: REV Journal on Electronics and Communications
volume: 5
number: 1--2
pages: 295--324
month: feb
monthnumber: 2
year: 2016
DOI: https://doi.org/10.21553/rev-jec.92
bibdate: Sat Dec 16 15:18:22 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2017.12.16 15:18:22 ???
URL: http://www.rev-jec.org/index.php/rev-jec/article/view/92
acknowledgement: ack-nhfb
journalURL: http://www.rev-jec.org/index.php/rev-jec/
rawauthor: Vo Nguyen Quoc Bao and Luu Pham Tuyen and Huynh Huu Tue
rawtitle: A Survey on Approximations of One-Dimensional {Gaussian} {$Q$}-Function
entry:
@Article{Bao:2016:SAO,
  author =       "Vo Nguyen Quoc Bao and Luu Pham Tuyen and Huynh Huu
                 Tue",
  title =        "A Survey on Approximations of One-Dimensional
                 {Gaussian} {$Q$}-Function",
  journal =      "{REV} Journal on Electronics and Communications",
  volume =       "5",
  number =       "1--2",
  month =        feb,
  year =         "2016",
  DOI =          "https://doi.org/10.21553/rev-jec.92",
  bibdate =      "Sat Dec 16 15:18:22 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.rev-jec.org/index.php/rev-jec/article/view/92",
  acknowledgement = ack-nhfb,
  journal-URL =  "http://www.rev-jec.org/index.php/rev-jec/",
}


bibtype: article
filename: elefunt.bib
label: Bytev:2016:HHF
author: Vladimir V. Bytev and Bernd A. Kniehl
journal: j-COMP-PHYS-COMM
volume: 206
pages: 78--83
month: sep
monthnumber: 9
year: 2016
CODEN: CPHCBZ
DOI: https://doi.org/10.1016/j.cpc.2016.04.016
ISSN: 0010-4655 (print), 1879-2944 (electronic) OR 00104655 (print), 18792944 (electronic)
ISSNL: 0010-4655 OR 00104655
bibdate: Fri Jun 10 18:27:25 MDT 2016
bibtimestamp: 2016.06.10 18:27:25 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0010465516301059
fjournal: Computer Physics Communications
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00104655/
rawauthor: Vladimir V. Bytev and Bernd A. Kniehl
title:
HYPERDIRE --- HYPERgeometric functions DIfferential REduction: Mathematica-based packages for the differential reduction of generalized hypergeometric functions: Lauricella function $ F_c $ of three variables

bibsource:
https://www.math.utah.edu/pub/tex/bib/compphyscomm2010.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathematica.bib

rawtitle:
{HYPERDIRE} --- {HYPERgeometric functions DIfferential REduction}: {Mathematica}-based packages for the differential reduction of generalized hypergeometric functions: {Lauricella} function {$ F_c $} of three variables

entry:
@Article{Bytev:2016:HHF,
  author =       "Vladimir V. Bytev and Bernd A. Kniehl",
  title =        "{HYPERDIRE} --- {HYPERgeometric functions DIfferential
                 REduction}: {Mathematica}-based packages for the
                 differential reduction of generalized hypergeometric
                 functions: {Lauricella} function {$ F_c $} of three
                 variables",
  journal =      j-COMP-PHYS-COMM,
  volume =       "206",
  number =       "??",
  pages =        "78--83",
  month =        sep,
  year =         "2016",
  CODEN =        "CPHCBZ",
  DOI =          "https://doi.org/10.1016/j.cpc.2016.04.016",
  ISSN =         "0010-4655 (print), 1879-2944 (electronic)",
  ISSN-L =       "0010-4655",
  bibdate =      "Fri Jun 10 18:27:25 MDT 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/compphyscomm2010.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/mathematica.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0010465516301059",
  acknowledgement = ack-nhfb,
  fjournal =     "Computer Physics Communications",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00104655/",
}


bibtype: article
filename: elefunt.bib
label: Chen:2016:AEG
author: Chao-Ping Chen
journal: j-APPL-MATH-COMP
volume: 276
pages: 417--431
day: 5
month: mar
monthnumber: 3
year: 2016
CODEN: AMHCBQ
ISSN: 0096-3003 (print), 1873-5649 (electronic) OR 00963003 (print), 18735649 (electronic)
ISSNL: 0096-3003 OR 00963003
bibdate: Tue Jan 26 17:22:21 MST 2016
bibtimestamp: 2016.01.26 17:22:21 MST
URL: http://www.sciencedirect.com/science/article/pii/S0096300315016057
fjournal: Applied Mathematics and Computation
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00963003/
rawauthor: Chao-Ping Chen
title:
On the asymptotic expansions of the gamma function related to the Nemes, Gosper and Burnside formulas

rawtitle:
On the asymptotic expansions of the gamma function related to the {Nemes}, {Gosper} and {Burnside} formulas

entry:
@Article{Chen:2016:AEG,
  author =       "Chao-Ping Chen",
  title =        "On the asymptotic expansions of the gamma function
                 related to the {Nemes}, {Gosper} and {Burnside}
                 formulas",
  journal =      j-APPL-MATH-COMP,
  volume =       "276",
  number =       "??",
  pages =        "417--431",
  day =          "5",
  month =        mar,
  year =         "2016",
  CODEN =        "AMHCBQ",
  ISSN =         "0096-3003 (print), 1873-5649 (electronic)",
  ISSN-L =       "0096-3003",
  bibdate =      "Tue Jan 26 17:22:21 MST 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/applmathcomput2015.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0096300315016057",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied Mathematics and Computation",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00963003/",
}


bibtype: article
filename: elefunt.bib
label: Chen:2016:IAEa
author: Chao-Ping Chen and Long Lin
title: Inequalities and asymptotic expansions for the gamma function related to Mortici's formula
journal: j-J-NUMBER-THEORY
volume: 162
pages: 578--588
month: may
monthnumber: 5
year: 2016
CODEN: JNUTA9
DOI: https://doi.org/10.1016/j.jnt.2015.09.014
ISSN: 0022-314X (print), 1096-1658 (electronic) OR 0022314X (print), 10961658 (electronic)
ISSNL: 0022-314X OR 0022314X
bibdate: Wed Jul 15 08:49:20 MDT 2020
bibtimestamp: 2020.07.15 08:49:20 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0022314X15003133
fjournal: Journal of Number Theory
acknowledgement: ack-nhfb
ajournal: J. Number Theory
journalURL: http://www.sciencedirect.com/science/journal/0022314X
rawauthor: Chao-Ping Chen and Long Lin
rawtitle: Inequalities and asymptotic expansions for the gamma function related to {Mortici}'s formula
entry:
@Article{Chen:2016:IAEa,
  author =       "Chao-Ping Chen and Long Lin",
  title =        "Inequalities and asymptotic expansions for the gamma
                 function related to {Mortici}'s formula",
  journal =      j-J-NUMBER-THEORY,
  volume =       "162",
  number =       "??",
  pages =        "578--588",
  month =        may,
  year =         "2016",
  CODEN =        "JNUTA9",
  DOI =          "https://doi.org/10.1016/j.jnt.2015.09.014",
  ISSN =         "0022-314X (print), 1096-1658 (electronic)",
  ISSN-L =       "0022-314X",
  bibdate =      "Wed Jul 15 08:49:20 MDT 2020",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jnumbertheory2010.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0022314X15003133",
  acknowledgement = ack-nhfb,
  ajournal =     "J. Number Theory",
  fjournal =     "Journal of Number Theory",
  journal-URL =  "http://www.sciencedirect.com/science/journal/0022314X",
}


bibtype: article
filename: elefunt.bib
label: Chen:2016:IAEb
author: Chao-Ping Chen
title: Inequalities and asymptotic expansions for the psi function and the Euler--Mascheroni constant
journal: j-J-NUMBER-THEORY
volume: 163
pages: 596--607
month: jun
monthnumber: 6
year: 2016
CODEN: JNUTA9
DOI: https://doi.org/10.1016/j.jnt.2015.10.013
ISSN: 0022-314X (print), 1096-1658 (electronic) OR 0022314X (print), 10961658 (electronic)
ISSNL: 0022-314X OR 0022314X
bibdate: Wed Jul 15 08:49:20 MDT 2020
bibtimestamp: 2020.07.15 08:49:20 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0022314X15003558
fjournal: Journal of Number Theory
acknowledgement: ack-nhfb
ajournal: J. Number Theory
journalURL: http://www.sciencedirect.com/science/journal/0022314X
rawauthor: Chao-Ping Chen
rawtitle: Inequalities and asymptotic expansions for the psi function and the {Euler--Mascheroni} constant
entry:
@Article{Chen:2016:IAEb,
  author =       "Chao-Ping Chen",
  title =        "Inequalities and asymptotic expansions for the psi
                 function and the {Euler--Mascheroni} constant",
  journal =      j-J-NUMBER-THEORY,
  volume =       "163",
  number =       "??",
  pages =        "596--607",
  month =        jun,
  year =         "2016",
  CODEN =        "JNUTA9",
  DOI =          "https://doi.org/10.1016/j.jnt.2015.10.013",
  ISSN =         "0022-314X (print), 1096-1658 (electronic)",
  ISSN-L =       "0022-314X",
  bibdate =      "Wed Jul 15 08:49:20 MDT 2020",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jnumbertheory2010.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0022314X15003558",
  acknowledgement = ack-nhfb,
  ajournal =     "J. Number Theory",
  fjournal =     "Journal of Number Theory",
  journal-URL =  "http://www.sciencedirect.com/science/journal/0022314X",
}


bibtype: article
filename: elefunt.bib
label: Chen:2016:IAEc
author: Chao-Ping Chen
title: Inequalities and asymptotics for the Euler--Mascheroni constant based on DeTemple's result
journal: j-NUMER-ALGORITHMS
volume: 73
number: 3
pages: 761--774
month: nov
monthnumber: 11
year: 2016
CODEN: NUALEG
DOI: https://doi.org/10.1007/s11075-016-0116-9
ISSN: 1017-1398 (print), 1572-9265 (electronic) OR 10171398 (print), 15729265 (electronic)
ISSNL: 1017-1398 OR 10171398
bibdate: Wed Mar 1 09:12:13 MST 2017
bibtimestamp: 2017.03.01 09:12:13 MST
URL: http://link.springer.com/article/10.1007/s11075-016-0116-9
fjournal: Numerical Algorithms
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11075
rawauthor: Chao-Ping Chen
rawtitle: Inequalities and asymptotics for the {Euler--Mascheroni} constant based on {DeTemple's} result
bibsource:
http://link.springer.com/journal/11075/73/3; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib

entry:
@Article{Chen:2016:IAEc,
  author =       "Chao-Ping Chen",
  title =        "Inequalities and asymptotics for the
                 {Euler--Mascheroni} constant based on {DeTemple's}
                 result",
  journal =      j-NUMER-ALGORITHMS,
  volume =       "73",
  number =       "3",
  pages =        "761--774",
  month =        nov,
  year =         "2016",
  CODEN =        "NUALEG",
  DOI =          "https://doi.org/10.1007/s11075-016-0116-9",
  ISSN =         "1017-1398 (print), 1572-9265 (electronic)",
  ISSN-L =       "1017-1398",
  bibdate =      "Wed Mar 1 09:12:13 MST 2017",
  bibsource =    "http://link.springer.com/journal/11075/73/3;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib",
  URL =          "http://link.springer.com/article/10.1007/s11075-016-0116-9",
  acknowledgement = ack-nhfb,
  fjournal =     "Numerical Algorithms",
  journal-URL =  "http://link.springer.com/journal/11075",
}


bibtype: article
filename: elefunt.bib
label: Chen:2016:MAA
author: Chao-Ping Chen
title: A more accurate approximation for the gamma function
journal: j-J-NUMBER-THEORY
volume: 164
pages: 417--428
month: jul
monthnumber: 7
year: 2016
CODEN: JNUTA9
DOI: https://doi.org/10.1016/j.jnt.2015.11.007
ISSN: 0022-314X (print), 1096-1658 (electronic) OR 0022314X (print), 10961658 (electronic)
ISSNL: 0022-314X OR 0022314X
bibdate: Wed Jul 15 08:49:21 MDT 2020
bibtimestamp: 2020.07.15 08:49:21 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0022314X16000068
fjournal: Journal of Number Theory
acknowledgement: ack-nhfb
ajournal: J. Number Theory
journalURL: http://www.sciencedirect.com/science/journal/0022314X
rawauthor: Chao-Ping Chen
rawtitle: A more accurate approximation for the gamma function
entry:
@Article{Chen:2016:MAA,
  author =       "Chao-Ping Chen",
  title =        "A more accurate approximation for the gamma function",
  journal =      j-J-NUMBER-THEORY,
  volume =       "164",
  number =       "??",
  pages =        "417--428",
  month =        jul,
  year =         "2016",
  CODEN =        "JNUTA9",
  DOI =          "https://doi.org/10.1016/j.jnt.2015.11.007",
  ISSN =         "0022-314X (print), 1096-1658 (electronic)",
  ISSN-L =       "0022-314X",
  bibdate =      "Wed Jul 15 08:49:21 MDT 2020",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jnumbertheory2010.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0022314X16000068",
  acknowledgement = ack-nhfb,
  ajournal =     "J. Number Theory",
  fjournal =     "Journal of Number Theory",
  journal-URL =  "http://www.sciencedirect.com/science/journal/0022314X",
}


bibtype: article
filename: elefunt.bib
label: Chen:2016:MPI
author: Chao-Ping Chen
title: Monotonicity properties, inequalities and asymptotic expansions associated with the gamma function
journal: j-APPL-MATH-COMP
volume: 283
pages: 385--396
day: 20
month: jun
monthnumber: 6
year: 2016
CODEN: AMHCBQ
ISSN: 0096-3003 (print), 1873-5649 (electronic) OR 00963003 (print), 18735649 (electronic)
ISSNL: 0096-3003 OR 00963003
bibdate: Tue Apr 5 07:51:07 MDT 2016
bibtimestamp: 2016.04.05 07:51:07 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0096300316301515
fjournal: Applied Mathematics and Computation
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00963003/
rawauthor: Chao-Ping Chen
rawtitle: Monotonicity properties, inequalities and asymptotic expansions associated with the gamma function
entry:
@Article{Chen:2016:MPI,
  author =       "Chao-Ping Chen",
  title =        "Monotonicity properties, inequalities and asymptotic
                 expansions associated with the gamma function",
  journal =      j-APPL-MATH-COMP,
  volume =       "283",
  number =       "??",
  pages =        "385--396",
  day =          "20",
  month =        jun,
  year =         "2016",
  CODEN =        "AMHCBQ",
  ISSN =         "0096-3003 (print), 1873-5649 (electronic)",
  ISSN-L =       "0096-3003",
  bibdate =      "Tue Apr 5 07:51:07 MDT 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/applmathcomput2015.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0096300316301515",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied Mathematics and Computation",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00963003/",
}


bibtype: article
filename: elefunt.bib
label: Chen:2016:SIAa
author: Chao-Ping Chen and Wei-Wei Tong
title: Sharp inequalities and asymptotic expansions for the gamma function
journal: j-J-NUMBER-THEORY
volume: 160
pages: 418--431
month: mar
monthnumber: 3
year: 2016
CODEN: JNUTA9
DOI: https://doi.org/10.1016/j.jnt.2015.09.021
ISSN: 0022-314X (print), 1096-1658 (electronic) OR 0022314X (print), 10961658 (electronic)
ISSNL: 0022-314X OR 0022314X
bibdate: Wed Jul 15 08:49:18 MDT 2020
bibtimestamp: 2020.07.15 08:49:18 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0022314X15003200
fjournal: Journal of Number Theory
acknowledgement: ack-nhfb
ajournal: J. Number Theory
journalURL: http://www.sciencedirect.com/science/journal/0022314X
rawauthor: Chao-Ping Chen and Wei-Wei Tong
rawtitle: Sharp inequalities and asymptotic expansions for the gamma function
entry:
@Article{Chen:2016:SIAa,
  author =       "Chao-Ping Chen and Wei-Wei Tong",
  title =        "Sharp inequalities and asymptotic expansions for the
                 gamma function",
  journal =      j-J-NUMBER-THEORY,
  volume =       "160",
  number =       "??",
  pages =        "418--431",
  month =        mar,
  year =         "2016",
  CODEN =        "JNUTA9",
  DOI =          "https://doi.org/10.1016/j.jnt.2015.09.021",
  ISSN =         "0022-314X (print), 1096-1658 (electronic)",
  ISSN-L =       "0022-314X",
  bibdate =      "Wed Jul 15 08:49:18 MDT 2020",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jnumbertheory2010.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0022314X15003200",
  acknowledgement = ack-nhfb,
  ajournal =     "J. Number Theory",
  fjournal =     "Journal of Number Theory",
  journal-URL =  "http://www.sciencedirect.com/science/journal/0022314X",
}


bibtype: article
filename: elefunt.bib
label: Erascu:2016:RQE
author: Madalina Erascu and Hoon Hong
journal: j-J-SYMBOLIC-COMP
volume: 75
pages: 110--126
month: jul
monthnumber: 7
year: 2016
CODEN: JSYCEH
ISSN: 0747-7171 (print), 1095-855X (electronic) OR 07477171 (print), 1095855X (electronic)
ISSNL: 0747-7171 OR 07477171
bibdate: Mon Jan 25 06:25:01 MST 2016
bibtimestamp: 2016.01.25 06:25:01 MST
URL: http://www.sciencedirect.com/science/article/pii/S0747717115001091
fjournal: Journal of Symbolic Computation
keywords: interval arithmetic; interval square root
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/07477171/
rawauthor: Madalina Erascu and Hoon Hong
title:
Real quantifier elimination for the synthesis of optimal numerical algorithms (Case study: Square root computation)

bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jsymcomp.bib

rawtitle:
Real quantifier elimination for the synthesis of optimal numerical algorithms (Case study: Square root computation)

entry:
@Article{Erascu:2016:RQE,
  author =       "Madalina Erascu and Hoon Hong",
  title =        "Real quantifier elimination for the synthesis of
                 optimal numerical algorithms (Case study: Square root
                 computation)",
  journal =      j-J-SYMBOLIC-COMP,
  volume =       "75",
  number =       "??",
  pages =        "110--126",
  month =        jul # "\slash " # aug,
  year =         "2016",
  CODEN =        "JSYCEH",
  ISSN =         "0747-7171 (print), 1095-855X (electronic)",
  ISSN-L =       "0747-7171",
  bibdate =      "Mon Jan 25 06:25:01 MST 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jsymcomp.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0747717115001091",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Symbolic Computation",
  journal-URL =  "http://www.sciencedirect.com/science/journal/07477171/",
  keywords =     "interval arithmetic; interval square root",
}


bibtype: techreport
filename: elefunt.bib
label: Fateman:2016:CUA
author: Richard J. Fateman
type: Report
institution: University of California, Berkeley
address: Berkeley, CA 947220-1776, USA
pages: 110--126
day: 4
month: jun
monthnumber: 6
year: 2016
bibdate: Fri Feb 24 09:55:02 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2017.02.24 09:55:02 ???
URL: https://people.eecs.berkeley.edu/~fateman/papers/hermite.pdf
acknowledgement: ack-nhfb
rawauthor: Richard J. Fateman
title:
Comments on Unrestricted Algorithms for Bessel Functions in Computer Algebra: Arbitrary Precision, The Backwards Recurrence, Taylor Series, Hermite Interpolation

abstract:
We explore various ways of implementing ``unrestricted algorithms'' [3] for approximating Bessel ($J$) functions. An unrestricted algorithm for a function $ f(x)$ provides a result to any requested precision in the answer. The emphasis is on higher-than-normal precision with the precision specified as an extra argument to the function. That is, the precision is specified at run-time. We require that the algorithm provide at least the requested number of correct digits, contrary to some existing codes which provide only ``absolute error'' near critical points. We use $ J_0 $ of real non-negative argument as an example, although much of the reasoning generalizes to other Bessel functions or related functions.par Since it is plausible that there will be requests for additional values of $ J_0 $ at the same (high) precision at a collection of nearby arguments, we consider implementations that cache certain re-usable key constants (namely zeros of $ J_0 $ near the argument values).

rawabstract:
We explore various ways of implementing ``unrestricted algorithms'' [3] for approximating Bessel ($J$) functions. An unrestricted algorithm for a function $ f(x)$ provides a result to any requested precision in the answer. The emphasis is on higher-than-normal precision with the precision specified as an extra argument to the function. That is, the precision is specified at run-time. We require that the algorithm provide at least the requested number of correct digits, contrary to some existing codes which provide only ``absolute error'' near critical points. We use $ J_0 $ of real non-negative argument as an example, although much of the reasoning generalizes to other Bessel functions or related functions.\par Since it is plausible that there will be requests for additional values of $ J_0 $ at the same (high) precision at a collection of nearby arguments, we consider implementations that cache certain re-usable key constants (namely zeros of $ J_0 $ near the argument values).

rawtitle:
Comments on Unrestricted Algorithms for {Bessel} Functions in Computer Algebra: Arbitrary Precision, The Backwards Recurrence, {Taylor} Series, {Hermite} Interpolation

entry:
@TechReport{Fateman:2016:CUA,
  author =       "Richard J. Fateman",
  title =        "Comments on Unrestricted Algorithms for {Bessel}
                 Functions in Computer Algebra: Arbitrary Precision, The
                 Backwards Recurrence, {Taylor} Series, {Hermite}
                 Interpolation",
  type =         "Report",
  institution =  "University of California, Berkeley",
  address =      "Berkeley, CA 947220-1776, USA",
  day =          "4",
  month =        jun,
  year =         "2016",
  bibdate =      "Fri Feb 24 09:55:02 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "https://people.eecs.berkeley.edu/~fateman/papers/hermite.pdf",
  abstract =     "We explore various ways of implementing ``unrestricted
                 algorithms'' [3] for approximating Bessel ($J$)
                 functions. An unrestricted algorithm for a function $
                 f(x)$ provides a result to any requested precision in
                 the answer. The emphasis is on higher-than-normal
                 precision with the precision specified as an extra
                 argument to the function. That is, the precision is
                 specified at run-time. We require that the algorithm
                 provide at least the requested number of correct
                 digits, contrary to some existing codes which provide
                 only ``absolute error'' near critical points. We use $
                 J_0 $ of real non-negative argument as an example,
                 although much of the reasoning generalizes to other
                 Bessel functions or related functions.\par

                 Since it is plausible that there will be requests for
                 additional values of $ J_0 $ at the same (high)
                 precision at a collection of nearby arguments, we
                 consider implementations that cache certain re-usable
                 key constants (namely zeros of $ J_0 $ near the
                 argument values).",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: elefunt.bib
label: Gautschi:2016:AER
author: Walter Gautschi
journal: j-TOMS
volume: 42
number: 1
pages: 9:1--9:10
month: feb
monthnumber: 2
year: 2016
CODEN: ACMSCU
DOI: https://doi.org/10.1145/2735626
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
bibdate: Tue Mar 1 17:07:56 MST 2016
bibtimestamp: 2016.03.01 17:07:56 MST
fjournal: ACM Transactions on Mathematical Software (TOMS)
acknowledgement: ack-nhfb
articleno: 9
journalURL: https://dl.acm.org/loi/toms
rawauthor: Walter Gautschi
title:
Algorithm 957: Evaluation of the Repeated Integral of the Coerror Function by Half-Range Gauss--Hermite Quadrature

bibsource:
http://www.acm.org/pubs/contents/journals/toms/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/matlab.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib

abstract:
Nonstandard Gaussian quadrature is applied to evaluate the repeated integral $ i^n erfc x $ of the coerror function for $ n in N_0 $, $ x in R $ in an appropriate domain of the $ (n, x)$-plane. Relevant software in MATLAB is provided: in particular, two routines evaluating the function to an accuracy of 12 respective 30-decimal digits.

rawabstract:
Nonstandard Gaussian quadrature is applied to evaluate the repeated integral $ i^n \erfc x $ of the coerror function for $ n \in N_0 $, $ x \in R $ in an appropriate domain of the $ (n, x)$-plane. Relevant software in MATLAB is provided: in particular, two routines evaluating the function to an accuracy of 12 respective 30-decimal digits.

rawtitle:
Algorithm 957: Evaluation of the Repeated Integral of the Coerror Function by Half-Range {Gauss--Hermite} Quadrature

entry:
@Article{Gautschi:2016:AER,
  author =       "Walter Gautschi",
  title =        "Algorithm 957: Evaluation of the Repeated Integral of
                 the Coerror Function by Half-Range {Gauss--Hermite}
                 Quadrature",
  journal =      j-TOMS,
  volume =       "42",
  number =       "1",
  pages =        "9:1--9:10",
  month =        feb,
  year =         "2016",
  CODEN =        "ACMSCU",
  DOI =          "https://doi.org/10.1145/2735626",
  ISSN =         "0098-3500 (print), 1557-7295 (electronic)",
  ISSN-L =       "0098-3500",
  bibdate =      "Tue Mar 1 17:07:56 MST 2016",
  bibsource =    "http://www.acm.org/pubs/contents/journals/toms/;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/matlab.bib;
                 https://www.math.utah.edu/pub/tex/bib/toms.bib",
  abstract =     "Nonstandard Gaussian quadrature is applied to evaluate
                 the repeated integral $ i^n \erfc x $ of the coerror
                 function for $ n \in N_0 $, $ x \in R $ in an
                 appropriate domain of the $ (n, x)$-plane. Relevant
                 software in MATLAB is provided: in particular, two
                 routines evaluating the function to an accuracy of 12
                 respective 30-decimal digits.",
  acknowledgement = ack-nhfb,
  articleno =    "9",
  fjournal =     "ACM Transactions on Mathematical Software (TOMS)",
  journal-URL =  "https://dl.acm.org/loi/toms",
}


bibtype: article
filename: elefunt.bib
label: Gil:2016:ACI
author: Amparo Gil and Diego Ruiz-Antolin and Javier Segura and Nico M. Temme
title: Algorithm 969: Computation of the Incomplete Gamma Function for Negative Values of the Argument
journal: j-TOMS
volume: 43
number: 3
pages: 26:1--26:9
month: nov
monthnumber: 11
year: 2016
CODEN: ACMSCU
DOI: https://doi.org/10.1145/2972951
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
bibdate: Tue Nov 22 17:45:25 MST 2016
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/toms.bib
bibtimestamp: 2016.11.22 17:45:25 MST
URL: http://dl.acm.org/citation.cfm?id=2972951
fjournal: ACM Transactions on Mathematical Software
acknowledgement: ack-nhfb
articleno: 26
journalURL: https://dl.acm.org/loi/toms
rawauthor: Amparo Gil and Diego Ruiz-Antol{\'\i}n and Javier Segura and Nico M. Temme
rawtitle: {Algorithm 969}: Computation of the Incomplete Gamma Function for Negative Values of the Argument
abstract:
An algorithm for computing the incomplete gamma function $ gamma ^(a, z) $ for real values of the parameter $a$ and negative real values of the argument $z$ is presented. The algorithm combines the use of series expansions, Poincare-type expansions, uniform asymptotic expansions, and recurrence relations, depending on the parameter region. A relative accuracy $ approx 10^-13$ in the parameter region $ (a, z) in [500, 500] times [500, 0)$ can be obtained when computing the function $ gamma ^ast (a, z)$ with the Fortran 90 module IncgamNEG implementing the algorithm.

rawabstract:
An algorithm for computing the incomplete gamma function $ \gamma^(a, z) $ for real values of the parameter $a$ and negative real values of the argument $z$ is presented. The algorithm combines the use of series expansions, Poincar{\'e}-type expansions, uniform asymptotic expansions, and recurrence relations, depending on the parameter region. A relative accuracy $ \approx 10^{-13}$ in the parameter region $ (a, z) \in [500, 500] \times [500, 0)$ can be obtained when computing the function $ \gamma^\ast (a, z)$ with the Fortran 90 module IncgamNEG implementing the algorithm.

entry:
@Article{Gil:2016:ACI,
  author =       "Amparo Gil and Diego Ruiz-Antol{\'\i}n and Javier
                 Segura and Nico M. Temme",
  title =        "{Algorithm 969}: Computation of the Incomplete Gamma
                 Function for Negative Values of the Argument",
  journal =      j-TOMS,
  volume =       "43",
  number =       "3",
  pages =        "26:1--26:9",
  month =        nov,
  year =         "2016",
  CODEN =        "ACMSCU",
  DOI =          "https://doi.org/10.1145/2972951",
  ISSN =         "0098-3500 (print), 1557-7295 (electronic)",
  ISSN-L =       "0098-3500",
  bibdate =      "Tue Nov 22 17:45:25 MST 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/toms.bib",
  URL =          "http://dl.acm.org/citation.cfm?id=2972951",
  abstract =     "An algorithm for computing the incomplete gamma
                 function $ \gamma^(a, z) $ for real values of the
                 parameter $a$ and negative real values of the argument
                 $z$ is presented. The algorithm combines the use of
                 series expansions, Poincar{\'e}-type expansions,
                 uniform asymptotic expansions, and recurrence
                 relations, depending on the parameter region. A
                 relative accuracy $ \approx 10^{-13}$ in the parameter
                 region $ (a, z) \in [500, 500] \times [500, 0)$ can be
                 obtained when computing the function $ \gamma^\ast (a,
                 z)$ with the Fortran 90 module IncgamNEG implementing
                 the algorithm.",
  acknowledgement = ack-nhfb,
  articleno =    "26",
  fjournal =     "ACM Transactions on Mathematical Software",
  journal-URL =  "https://dl.acm.org/loi/toms",
}


bibtype: article
filename: elefunt.bib
label: Giles:2016:AAI
author: Michael B. Giles
title: Algorithm 955: Approximation of the Inverse Poisson Cumulative Distribution Function
journal: j-TOMS
volume: 42
number: 1
pages: 7:1--7:22
month: feb
monthnumber: 2
year: 2016
CODEN: ACMSCU
DOI: https://doi.org/10.1145/2699466
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
bibdate: Tue Mar 1 17:07:56 MST 2016
bibtimestamp: 2016.03.01 17:07:56 MST
fjournal: ACM Transactions on Mathematical Software (TOMS)
acknowledgement: ack-nhfb
articleno: 7
journalURL: https://dl.acm.org/loi/toms
rawauthor: Michael B. Giles
rawtitle: Algorithm 955: Approximation of the Inverse {Poisson} Cumulative Distribution Function
abstract:
New approximations for the inverse of the incomplete gamma function are derived, which are used to develop efficient evaluations of the inverse Poisson cumulative distribution function. An asymptotic approximation based on the standard Normal approximation is particularly good for CPUs with MIMD cores, while for GPUs and other hardware with vector units, a second asymptotic approximation based on Temme's approximation of the incomplete gamma function is more efficient due to conditional branching within each vector. The accuracy and efficiency of the software implementations is assessed on both CPUs and GPUs.

rawabstract:
New approximations for the inverse of the incomplete gamma function are derived, which are used to develop efficient evaluations of the inverse Poisson cumulative distribution function. An asymptotic approximation based on the standard Normal approximation is particularly good for CPUs with MIMD cores, while for GPUs and other hardware with vector units, a second asymptotic approximation based on Temme's approximation of the incomplete gamma function is more efficient due to conditional branching within each vector. The accuracy and efficiency of the software implementations is assessed on both CPUs and GPUs.

entry:
@Article{Giles:2016:AAI,
  author =       "Michael B. Giles",
  title =        "Algorithm 955: Approximation of the Inverse {Poisson}
                 Cumulative Distribution Function",
  journal =      j-TOMS,
  volume =       "42",
  number =       "1",
  pages =        "7:1--7:22",
  month =        feb,
  year =         "2016",
  CODEN =        "ACMSCU",
  DOI =          "https://doi.org/10.1145/2699466",
  ISSN =         "0098-3500 (print), 1557-7295 (electronic)",
  ISSN-L =       "0098-3500",
  bibdate =      "Tue Mar 1 17:07:56 MST 2016",
  bibsource =    "http://www.acm.org/pubs/contents/journals/toms/;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/toms.bib",
  abstract =     "New approximations for the inverse of the incomplete
                 gamma function are derived, which are used to develop
                 efficient evaluations of the inverse Poisson cumulative
                 distribution function. An asymptotic approximation
                 based on the standard Normal approximation is
                 particularly good for CPUs with MIMD cores, while for
                 GPUs and other hardware with vector units, a second
                 asymptotic approximation based on Temme's approximation
                 of the incomplete gamma function is more efficient due
                 to conditional branching within each vector. The
                 accuracy and efficiency of the software implementations
                 is assessed on both CPUs and GPUs.",
  acknowledgement = ack-nhfb,
  articleno =    "7",
  fjournal =     "ACM Transactions on Mathematical Software (TOMS)",
  journal-URL =  "https://dl.acm.org/loi/toms",
}


bibtype: article
filename: elefunt.bib
label: Jameson:2016:IGF
author: G. J. O. Jameson
title: The incomplete gamma functions
journal: j-MATH-GAZ
volume: 100
number: 548
pages: 298--306
month: jul
monthnumber: 7
year: 2016
CODEN: MAGAAS
DOI: https://doi.org/10.1017/mag.2016.67
ISSN: 0025-5572 (print), 2056-6328 (electronic) OR 00255572 (print), 20566328 (electronic)
ISSNL: 0025-5572 OR 00255572
bibdate: Tue Sep 27 10:11:13 MDT 2016
bibtimestamp: 2016.09.27 10:11:13 MDT
URL: https://www.cambridge.org/core/product/9373A31AD28D793AB5431E35EA5C2AF6
fjournal: The Mathematical Gazette
acknowledgement: ack-nhfb
ajournal: Math. Gaz.
onlinedate: 14 June 2016
rawauthor: G. J. O. Jameson
rawtitle: The incomplete gamma functions
bibsource:
http://journals.cambridge.org/action/displayIssue?jid=MAG; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathgaz2010.bib

remark:
This paper exhibits and proves several useful identities for the incomplete gamma functions, but does not discuss their stable numerical computation.

journalURL:
http://journals.cambridge.org/action/displayIssue?jid=MAG; http://www.m-a.org.uk/jsp/index.jsp?lnk=620

entry:
@Article{Jameson:2016:IGF,
  author =       "G. J. O. Jameson",
  title =        "The incomplete gamma functions",
  journal =      j-MATH-GAZ,
  volume =       "100",
  number =       "548",
  pages =        "298--306",
  month =        jul,
  year =         "2016",
  CODEN =        "MAGAAS",
  DOI =          "https://doi.org/10.1017/mag.2016.67",
  ISSN =         "0025-5572 (print), 2056-6328 (electronic)",
  ISSN-L =       "0025-5572",
  bibdate =      "Tue Sep 27 10:11:13 MDT 2016",
  bibsource =    "http://journals.cambridge.org/action/displayIssue?jid=MAG;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/mathgaz2010.bib",
  URL =          "https://www.cambridge.org/core/product/9373A31AD28D793AB5431E35EA5C2AF6",
  acknowledgement = ack-nhfb,
  ajournal =     "Math. Gaz.",
  fjournal =     "The Mathematical Gazette",
  journal-URL =  "http://journals.cambridge.org/action/displayIssue?jid=MAG;
                 http://www.m-a.org.uk/jsp/index.jsp?lnk=620",
  onlinedate =   "14 June 2016",
  remark =       "This paper exhibits and proves several useful
                 identities for the incomplete gamma functions, but does
                 not discuss their stable numerical computation.",
}


bibtype: article
filename: elefunt.bib
label: Johansson:2016:CHF
author: Fredrik Johansson
title: Computing hypergeometric functions rigorously
journal: arxiv.org
pages: 2--29
day: 22
month: jun
monthnumber: 6
year: 2016
bibdate: Thu Jun 23 07:39:32 2016
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2016.06.23 07:39:32 ???
URL: http://arxiv.org/abs/1606.06977
acknowledgement: ack-nhfb
rawauthor: Fredrik Johansson
rawtitle: Computing hypergeometric functions rigorously
abstract:
We present an efficient implementation of hypergeometric functions in arbitrary-precision interval arithmetic. The functions 0F1, 1F1, 2F1 and 2F0 (or the Kummer U-function) are supported for unrestricted complex parameters and argument, and by extension, we cover exponential and trigonometric integrals, error functions, Fresnel integrals, incomplete gamma and beta functions, Bessel functions, Airy functions, Legendre functions, Jacobi polynomials, complete elliptic integrals, and other special functions. The output can be used directly for interval computations or to generate provably correct floating-point approximations in any format. Performance is competitive with earlier arbitrary-precision software, and sometimes orders of magnitude faster. We also partially cover the generalized hypergeometric function pFq and computation of high-order parameter derivatives.

rawabstract:
We present an efficient implementation of hypergeometric functions in arbitrary-precision interval arithmetic. The functions 0F1, 1F1, 2F1 and 2F0 (or the Kummer U-function) are supported for unrestricted complex parameters and argument, and by extension, we cover exponential and trigonometric integrals, error functions, Fresnel integrals, incomplete gamma and beta functions, Bessel functions, Airy functions, Legendre functions, Jacobi polynomials, complete elliptic integrals, and other special functions. The output can be used directly for interval computations or to generate provably correct floating-point approximations in any format. Performance is competitive with earlier arbitrary-precision software, and sometimes orders of magnitude faster. We also partially cover the generalized hypergeometric function pFq and computation of high-order parameter derivatives.

entry:
@Article{Johansson:2016:CHF,
  author =       "Fredrik Johansson",
  title =        "Computing hypergeometric functions rigorously",
  journal =      "arxiv.org",
  volume =       "??",
  number =       "??",
  pages =        "2--29",
  day =          "22",
  month =        jun,
  year =         "2016",
  bibdate =      "Thu Jun 23 07:39:32 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://arxiv.org/abs/1606.06977",
  abstract =     "We present an efficient implementation of
                 hypergeometric functions in arbitrary-precision
                 interval arithmetic. The functions 0F1, 1F1, 2F1 and
                 2F0 (or the Kummer U-function) are supported for
                 unrestricted complex parameters and argument, and by
                 extension, we cover exponential and trigonometric
                 integrals, error functions, Fresnel integrals,
                 incomplete gamma and beta functions, Bessel functions,
                 Airy functions, Legendre functions, Jacobi polynomials,
                 complete elliptic integrals, and other special
                 functions. The output can be used directly for interval
                 computations or to generate provably correct
                 floating-point approximations in any format.
                 Performance is competitive with earlier
                 arbitrary-precision software, and sometimes orders of
                 magnitude faster. We also partially cover the
                 generalized hypergeometric function pFq and computation
                 of high-order parameter derivatives.",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: elefunt.bib
label: Johansson:2016:FAE
author: H. T. Johansson and C. Forssen
journal: j-SIAM-J-SCI-COMP
volume: 38
number: 1
pages: A376--A384
year: 2016
CODEN: SJOCE3
DOI: https://doi.org/10.1137/15M1021908
ISSN: 1064-8275 (print), 1095-7197 (electronic) OR 10648275 (print), 10957197 (electronic)
ISSNL: 1064-8275 OR 10648275
bibdate: Tue Jun 21 08:11:55 MDT 2016
bibtimestamp: 2016.06.21 08:11:55 MDT
fjournal: SIAM Journal on Scientific Computing
acknowledgement: ack-nhfb
journalURL: http://epubs.siam.org/sisc
onlinedate: January 2016
rawauthor: H. T. Johansson and C. Forss{\'e}n
title:
Fast and Accurate Evaluation of Wigner 3$j$, 6$j$, and 9$j$ Symbols Using Prime Factorization and Multiword Integer Arithmetic

bibsource:
http://epubs.siam.org/toc/sjoce3/38/1; https://www.math.utah.edu/pub/bibnet/authors/w/wigner-eugene.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/siamjscicomput.bib

rawtitle:
Fast and Accurate Evaluation of {Wigner} 3$j$, 6$j$, and 9$j$ Symbols Using Prime Factorization and Multiword Integer Arithmetic

entry:
@Article{Johansson:2016:FAE,
  author =       "H. T. Johansson and C. Forss{\'e}n",
  title =        "Fast and Accurate Evaluation of {Wigner} 3$j$, 6$j$,
                 and 9$j$ Symbols Using Prime Factorization and
                 Multiword Integer Arithmetic",
  journal =      j-SIAM-J-SCI-COMP,
  volume =       "38",
  number =       "1",
  pages =        "A376--A384",
  month =        "????",
  year =         "2016",
  CODEN =        "SJOCE3",
  DOI =          "https://doi.org/10.1137/15M1021908",
  ISSN =         "1064-8275 (print), 1095-7197 (electronic)",
  ISSN-L =       "1064-8275",
  bibdate =      "Tue Jun 21 08:11:55 MDT 2016",
  bibsource =    "http://epubs.siam.org/toc/sjoce3/38/1;
                 https://www.math.utah.edu/pub/bibnet/authors/w/wigner-eugene.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/siamjscicomput.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "SIAM Journal on Scientific Computing",
  journal-URL =  "http://epubs.siam.org/sisc",
  onlinedate =   "January 2016",
}


bibtype: article
filename: elefunt.bib
label: Koelink:2016:AST
author: Erik Koelink
title: Applications of spectral theory to special functions
journal: ArXiv e-prints
pages: 1--63
month: dec
monthnumber: 12
year: 2016
bibdate: Sat Feb 18 09:23:20 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2017.02.18 09:23:20 ???
URL: https://arxiv.org/abs/1612.07035
keywords: Mathematics --- Classical Analysis and ODEs; Mathematics --- Functional Analysis
acknowledgement: ack-nhfb
rawauthor: Erik Koelink
rawtitle: Applications of spectral theory to special functions
abstract:
Many special functions are eigenfunctions to explicit operators, such as difference and differential operators, which is in particular true for the special functions occurring in the Askey-scheme, its $q$-analogue and extensions. The study of the spectral properties of such operators leads to explicit information for the corresponding special functions. We discuss several instances of this application, involving orthogonal polynomials and their matrix-valued analogues.

rawabstract:
Many special functions are eigenfunctions to explicit operators, such as difference and differential operators, which is in particular true for the special functions occurring in the Askey-scheme, its $q$-analogue and extensions. The study of the spectral properties of such operators leads to explicit information for the corresponding special functions. We discuss several instances of this application, involving orthogonal polynomials and their matrix-valued analogues.

entry:
@Article{Koelink:2016:AST,
  author =       "Erik Koelink",
  title =        "Applications of spectral theory to special functions",
  journal =      "ArXiv e-prints",
  volume =       "??",
  pages =        "1--63",
  month =        dec,
  year =         "2016",
  bibdate =      "Sat Feb 18 09:23:20 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "https://arxiv.org/abs/1612.07035",
  abstract =     "Many special functions are eigenfunctions to explicit
                 operators, such as difference and differential
                 operators, which is in particular true for the special
                 functions occurring in the Askey-scheme, its
                 $q$-analogue and extensions. The study of the spectral
                 properties of such operators leads to explicit
                 information for the corresponding special functions. We
                 discuss several instances of this application,
                 involving orthogonal polynomials and their
                 matrix-valued analogues.",
  acknowledgement = ack-nhfb,
  eprint =       "1612.07035",
  keywords =     "Mathematics --- Classical Analysis and ODEs;
                 Mathematics --- Functional Analysis",
  primaryclass = "math.CA",
}


bibtype: article
filename: elefunt.bib
label: Kutsuna:2016:ARM
author: Takuro Kutsuna and Yoshinao Ishii
title: Abstraction and refinement of mathematical functions toward SMT-based test-case generation
journal: j-INT-J-SOFTW-TOOLS-TECHNOL-TRANSFER
volume: 18
number: 1
pages: 109--120
month: feb
monthnumber: 2
year: 2016
DOI: https://doi.org/10.1007/s10009-015-0389-7
ISSN: 1433-2779 (print), 1433-2787 (electronic) OR 14332779 (print), 14332787 (electronic)
ISSNL: 1433-2779 OR 14332779
bibdate: Mon Jan 25 08:12:53 MST 2016
bibtimestamp: 2016.01.25 08:12:53 MST
URL: http://link.springer.com/article/10.1007/s10009-015-0389-7
fjournal: International Journal on Software Tools for Technology Transfer (STTT)
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/10009
rawauthor: Takuro Kutsuna and Yoshinao Ishii
rawtitle: Abstraction and refinement of mathematical functions toward {SMT}-based test-case generation
bibsource:
http://link.springer.com/journal/10009/18/1; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/multithreading.bib; https://www.math.utah.edu/pub/tex/bib/sttt.bib

entry:
@Article{Kutsuna:2016:ARM,
  author =       "Takuro Kutsuna and Yoshinao Ishii",
  title =        "Abstraction and refinement of mathematical functions
                 toward {SMT}-based test-case generation",
  journal =      j-INT-J-SOFTW-TOOLS-TECHNOL-TRANSFER,
  volume =       "18",
  number =       "1",
  pages =        "109--120",
  month =        feb,
  year =         "2016",
  CODEN =        "????",
  DOI =          "https://doi.org/10.1007/s10009-015-0389-7",
  ISSN =         "1433-2779 (print), 1433-2787 (electronic)",
  ISSN-L =       "1433-2779",
  bibdate =      "Mon Jan 25 08:12:53 MST 2016",
  bibsource =    "http://link.springer.com/journal/10009/18/1;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/multithreading.bib;
                 https://www.math.utah.edu/pub/tex/bib/sttt.bib",
  URL =          "http://link.springer.com/article/10.1007/s10009-015-0389-7",
  acknowledgement = ack-nhfb,
  fjournal =     "International Journal on Software Tools for Technology
                 Transfer (STTT)",
  journal-URL =  "http://link.springer.com/journal/10009",
}


bibtype: inproceedings
filename: elefunt.bib
label: Langhammer:2016:SPN
author: Martin Langhammer and Bogdan Pasca
title: Single Precision Natural Logarithm Architecture for Hard Floating-Point and DSP-Enabled FPGAs
crossref: Montuschi:2016:ISC
pages: 164--171
year: 2016
DOI: https://doi.org/10.1109/ARITH.2016.20
bibdate: Fri Dec 16 15:17:20 MST 2016
bibtimestamp: 2016.12.16 15:17:20 MST
keywords: ARITH-23
acknowledgement: ack-nhfb
rawauthor: Martin Langhammer and Bogdan Pasca
rawtitle: Single Precision Natural Logarithm Architecture for Hard Floating-Point and {DSP}-Enabled {FPGAs}
entry:
@InProceedings{Langhammer:2016:SPN,
  author =       "Martin Langhammer and Bogdan Pasca",
  title =        "Single Precision Natural Logarithm Architecture for
                 Hard Floating-Point and {DSP}-Enabled {FPGAs}",
  crossref =     "Montuschi:2016:ISC",
  pages =        "164--171",
  year =         "2016",
  DOI =          "https://doi.org/10.1109/ARITH.2016.20",
  bibdate =      "Fri Dec 16 15:17:20 MST 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH-23",
}


bibtype: inproceedings
filename: elefunt.bib
label: LeMaire:2016:CFP
author: Julien Le Maire and Nicolas Brunie and Florent de Dinechin and Jean-Michel Muller
title: Computing floating-point logarithms with fixed-point operations
crossref: Montuschi:2016:ISC
pages: 156--163
year: 2016
DOI: https://doi.org/10.1109/ARITH.2016.24
bibdate: Fri Dec 16 15:17:20 MST 2016
bibtimestamp: 2016.12.16 15:17:20 MST
keywords: ARITH-23
acknowledgement: ack-nhfb
rawauthor: Julien {Le Maire} and Nicolas Brunie and Florent de Dinechin and Jean-Michel Muller
rawtitle: Computing floating-point logarithms with fixed-point operations
entry:
@InProceedings{LeMaire:2016:CFP,
  author =       "Julien {Le Maire} and Nicolas Brunie and Florent de
                 Dinechin and Jean-Michel Muller",
  title =        "Computing floating-point logarithms with fixed-point
                 operations",
  crossref =     "Montuschi:2016:ISC",
  pages =        "156--163",
  year =         "2016",
  DOI =          "https://doi.org/10.1109/ARITH.2016.24",
  bibdate =      "Fri Dec 16 15:17:20 MST 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH-23",
}


bibtype: article
filename: elefunt.bib
label: Lu:2016:QCF
author: Dawei Lu and Lixin Song and Congxu Ma
journal: j-NUMER-ALGORITHMS
volume: 72
number: 4
pages: 865--874
month: aug
monthnumber: 8
year: 2016
CODEN: NUALEG
DOI: https://doi.org/10.1007/s11075-015-0070-y
ISSN: 1017-1398 (print), 1572-9265 (electronic) OR 10171398 (print), 15729265 (electronic)
ISSNL: 1017-1398 OR 10171398
bibdate: Tue Sep 20 10:57:47 MDT 2016
bibtimestamp: 2016.09.20 10:57:47 MDT
URL: http://link.springer.com/article/10.1007/s11075-015-0070-y
fjournal: Numerical Algorithms
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11075
rawauthor: Dawei Lu and Lixin Song and Congxu Ma
title:
A quicker continued fraction approximation of the gamma function related to the Windschitl's formula

bibsource:
http://link.springer.com/journal/11075/72/4; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib

remark:
The tables at the end of this paper compare six algorithms for approximating $ n! $ for $ n = 50, 100, 500, 2500 $. The Burnside, Nemes, and Windschitl formulas are slightly less accurate than the traditional Stirling approximation. The new formula, and the Mortici formula, are slightly better than Stirling's.

rawtitle:
A quicker continued fraction approximation of the gamma function related to the {Windschitl}'s formula

entry:
@Article{Lu:2016:QCF,
  author =       "Dawei Lu and Lixin Song and Congxu Ma",
  title =        "A quicker continued fraction approximation of the
                 gamma function related to the {Windschitl}'s formula",
  journal =      j-NUMER-ALGORITHMS,
  volume =       "72",
  number =       "4",
  pages =        "865--874",
  month =        aug,
  year =         "2016",
  CODEN =        "NUALEG",
  DOI =          "https://doi.org/10.1007/s11075-015-0070-y",
  ISSN =         "1017-1398 (print), 1572-9265 (electronic)",
  ISSN-L =       "1017-1398",
  bibdate =      "Tue Sep 20 10:57:47 MDT 2016",
  bibsource =    "http://link.springer.com/journal/11075/72/4;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib",
  URL =          "http://link.springer.com/article/10.1007/s11075-015-0070-y",
  acknowledgement = ack-nhfb,
  fjournal =     "Numerical Algorithms",
  journal-URL =  "http://link.springer.com/journal/11075",
  remark =       "The tables at the end of this paper compare six
                 algorithms for approximating $ n! $ for $ n = 50, 100,
                 500, 2500 $. The Burnside, Nemes, and Windschitl
                 formulas are slightly less accurate than the
                 traditional Stirling approximation. The new formula,
                 and the Mortici formula, are slightly better than
                 Stirling's.",
}


bibtype: article
filename: elefunt.bib
label: Maignan:2016:FGL
author: Aude Maignan and Tony C. Scott
title: Fleshing out the generalized Lambert $W$ function
journal: j-ACM-COMM-COMP-ALGEBRA
volume: 50
number: 2
pages: 45--60
month: jun
monthnumber: 6
year: 2016
DOI: https://doi.org/10.1145/2992274.2992275
ISSN: 1932-2232 (print), 1932-2240 (electronic) OR 19322232 (print), 19322240 (electronic)
ISSNL: 1932-2232 OR 19322232
bibdate: Thu Aug 25 17:57:39 MDT 2016
bibtimestamp: 2016.08.25 17:57:39 MDT
fjournal: ACM Communications in Computer Algebra
acknowledgement: ack-nhfb
journalURL: http://portal.acm.org/browse_dl.cfm?idx=J1000
rawauthor: Aude Maignan and Tony C. Scott
rawtitle: Fleshing out the generalized {Lambert} {$W$} function
bibsource:
http://portal.acm.org/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/sigsam.bib

abstract:
Herein, we use Hardy's notion of the ``false derivative'' to obtain exact multiple roots in closed form of the transcendental--algebraic equations representing the generalized Lambert $W$ function. In this fashion, we flesh out the generalized Lambert $W$ function by complementing previous developments to produce a more complete and integrated body of work. Finally, we demonstrate the usefulness of this special function with some applications.

rawabstract:
Herein, we use Hardy's notion of the ``false derivative'' to obtain exact multiple roots in closed form of the transcendental--algebraic equations representing the generalized Lambert $W$ function. In this fashion, we flesh out the generalized Lambert $W$ function by complementing previous developments to produce a more complete and integrated body of work. Finally, we demonstrate the usefulness of this special function with some applications.

entry:
@Article{Maignan:2016:FGL,
  author =       "Aude Maignan and Tony C. Scott",
  title =        "Fleshing out the generalized {Lambert} {$W$}
                 function",
  journal =      j-ACM-COMM-COMP-ALGEBRA,
  volume =       "50",
  number =       "2",
  pages =        "45--60",
  month =        jun,
  year =         "2016",
  CODEN =        "????",
  DOI =          "https://doi.org/10.1145/2992274.2992275",
  ISSN =         "1932-2232 (print), 1932-2240 (electronic)",
  ISSN-L =       "1932-2232",
  bibdate =      "Thu Aug 25 17:57:39 MDT 2016",
  bibsource =    "http://portal.acm.org/;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/sigsam.bib",
  abstract =     "Herein, we use Hardy's notion of the ``false
                 derivative'' to obtain exact multiple roots in closed
                 form of the transcendental--algebraic equations
                 representing the generalized Lambert $W$ function. In
                 this fashion, we flesh out the generalized Lambert $W$
                 function by complementing previous developments to
                 produce a more complete and integrated body of work.
                 Finally, we demonstrate the usefulness of this special
                 function with some applications.",
  acknowledgement = ack-nhfb,
  fjournal =     "ACM Communications in Computer Algebra",
  journal-URL =  "http://portal.acm.org/browse_dl.cfm?idx=J1000",
}


bibtype: article
filename: elefunt.bib
label: Martin-Dorel:2016:PTB
author: Erik Martin-Dorel and Guillaume Melquiond
title: Proving Tight Bounds on Univariate Expressions with Elementary Functions in Coq
journal: j-J-AUTOM-REASON
volume: 57
number: 3
pages: 187--217
month: oct
monthnumber: 10
year: 2016
CODEN: JAREEW
DOI: https://doi.org/10.1007/s10817-015-9350-4
ISSN: 0168-7433 (print), 1573-0670 (electronic) OR 01687433 (print), 15730670 (electronic)
ISSNL: 0168-7433 OR 01687433
bibdate: Fri Sep 2 06:39:36 MDT 2016
bibtimestamp: 2016.09.02 06:39:36 MDT
URL: http://link.springer.com/accesspage/article/10.1007/s10817-015-9350-4
fjournal: Journal of Automated Reasoning
acknowledgement: ack-nhfb
ajournal: J. Autom. Reason.
journalURL: http://link.springer.com/journal/10817
rawauthor: {\'E}rik Martin-Dorel and Guillaume Melquiond
rawtitle: Proving Tight Bounds on Univariate Expressions with Elementary Functions in {Coq}
bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/jautomreason.bib

entry:
@Article{Martin-Dorel:2016:PTB,
  author =       "{\'E}rik Martin-Dorel and Guillaume Melquiond",
  title =        "Proving Tight Bounds on Univariate Expressions with
                 Elementary Functions in {Coq}",
  journal =      j-J-AUTOM-REASON,
  volume =       "57",
  number =       "3",
  pages =        "187--217",
  month =        oct,
  year =         "2016",
  CODEN =        "JAREEW",
  DOI =          "https://doi.org/10.1007/s10817-015-9350-4",
  ISSN =         "0168-7433 (print), 1573-0670 (electronic)",
  ISSN-L =       "0168-7433",
  bibdate =      "Fri Sep 2 06:39:36 MDT 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/jautomreason.bib",
  URL =          "http://link.springer.com/accesspage/article/10.1007/s10817-015-9350-4",
  acknowledgement = ack-nhfb,
  ajournal =     "J. Autom. Reason.",
  fjournal =     "Journal of Automated Reasoning",
  journal-URL =  "http://link.springer.com/journal/10817",
}


bibtype: article
filename: elefunt.bib
label: Mohankumar:2016:VAN
author: N. Mohankumar and A. Natarajan
title: On the very accurate numerical evaluation of the Generalized Fermi--Dirac Integrals
journal: j-COMP-PHYS-COMM
volume: 207
pages: 193--201
month: oct
monthnumber: 10
year: 2016
CODEN: CPHCBZ
ISSN: 0010-4655 (print), 1879-2944 (electronic) OR 00104655 (print), 18792944 (electronic)
ISSNL: 0010-4655 OR 00104655
bibdate: Tue Aug 30 18:08:51 MDT 2016
bibtimestamp: 2016.08.30 18:08:51 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0010465516301667
fjournal: Computer Physics Communications
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00104655/
rawauthor: N. Mohankumar and A. Natarajan
rawtitle: On the very accurate numerical evaluation of the {Generalized Fermi--Dirac Integrals}
entry:
@Article{Mohankumar:2016:VAN,
  author =       "N. Mohankumar and A. Natarajan",
  title =        "On the very accurate numerical evaluation of the
                 {Generalized Fermi--Dirac Integrals}",
  journal =      j-COMP-PHYS-COMM,
  volume =       "207",
  number =       "??",
  pages =        "193--201",
  month =        oct,
  year =         "2016",
  CODEN =        "CPHCBZ",
  ISSN =         "0010-4655 (print), 1879-2944 (electronic)",
  ISSN-L =       "0010-4655",
  bibdate =      "Tue Aug 30 18:08:51 MDT 2016",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/compphyscomm2010.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0010465516301667",
  acknowledgement = ack-nhfb,
  fjournal =     "Computer Physics Communications",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00104655/",
}


bibtype: article
filename: elefunt.bib
label: Moroz:2016:FCI
title: Fast calculation of inverse square root with the use of magic constant --- analytical approach
journal: arXiv.org
pages: 1--23
day: 14
month: mar
monthnumber: 3
year: 2016
DOI: https://doi.org/10.48550/arXiv.1603.04483
bibdate: Wed Dec 20 07:34:12 2023
bibtimestamp: 2023.12.20 07:34:12 ???
URL: https://arxiv.org/pdf/1603.04483.pdf
acknowledgement: ack-nhfb
rawtitle: Fast calculation of inverse square root with the use of magic constant --- analytical approach
author:
Leonid V. Moroz and Cezary J. Walczyk and Andriy Hrynchyshyn and Vijay Holimath and Jan L. Cieslinski

abstract:
We present a mathematical analysis of transformations used in fast calculation of inverse square root for single-precision floating-point numbers. Optimal values of the so called magic constants are derived in a systematic way, minimizing either absolute or relative errors at subsequent stages of the discussed algorithm.

rawabstract:
We present a mathematical analysis of transformations used in fast calculation of inverse square root for single-precision floating-point numbers. Optimal values of the so called magic constants are derived in a systematic way, minimizing either absolute or relative errors at subsequent stages of the discussed algorithm.

rawauthor:
Leonid V. Moroz and Cezary J. Walczyk and Andriy Hrynchyshyn and Vijay Holimath and Jan L. Cie{\'s}li{\'n}ski

entry:
@Article{Moroz:2016:FCI,
  author =       "Leonid V. Moroz and Cezary J. Walczyk and Andriy
                 Hrynchyshyn and Vijay Holimath and Jan L.
                 Cie{\'s}li{\'n}ski",
  title =        "Fast calculation of inverse square root with the use
                 of magic constant --- analytical approach",
  journal =      "arXiv.org",
  volume =       "??",
  number =       "??",
  pages =        "1--23",
  day =          "14",
  month =        mar,
  year =         "2016",
  DOI =          "https://doi.org/10.48550/arXiv.1603.04483",
  bibdate =      "Wed Dec 20 07:34:12 2023",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "https://arxiv.org/pdf/1603.04483.pdf",
  abstract =     "We present a mathematical analysis of transformations
                 used in fast calculation of inverse square root for
                 single-precision floating-point numbers. Optimal values
                 of the so called magic constants are derived in a
                 systematic way, minimizing either absolute or relative
                 errors at subsequent stages of the discussed
                 algorithm.",
  acknowledgement = ack-nhfb,
}


bibtype: misc
filename: elefunt.bib
label: Munshi:2016:OCS
author: Aaftab Munshi and Lee Howes and Bartosz Sochacki and Khronos OpenCL Working Group
title: The OpenCL C Specification Version: 2.0 Document Revision: 33
pages: 205
day: 13
month: apr
monthnumber: 4
year: 2016
bibdate: Mon Apr 16 14:05:49 2018
bibtimestamp: 2018.04.16 14:05:49 ???
URL: https://www.khronos.org/registry/OpenCL/specs/opencl-2.0-openclc.pdf
acknowledgement: ack-nhfb
howpublished: Web document.
rawauthor: Aaftab Munshi and Lee Howes and Bartosz Sochacki and {Khronos OpenCL Working Group}
rawtitle: The {OpenCL} {C} Specification Version: 2.0 Document Revision: 33
bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib; https://www.math.utah.edu/pub/tex/bib/pvm.bib

remark:
Section 6.1.3.2 Math Functions, pages 74ff, defines a function repertoire extended beyond that of ISO C, including tt acospi, tt asinpi, tt atanpi, tt atan2pi, tt cospi, tt sinpi, tt tanpi, tt cospi, tt fract, tt lgamma_r, tt mad (approximation to tt a * b + c), tt minmag, tt pown, tt rootn, tt sincos, tt sinpi, and tt tanpi.

entry:
@Misc{Munshi:2016:OCS,
  author =       "Aaftab Munshi and Lee Howes and Bartosz Sochacki and
                 {Khronos OpenCL Working Group}",
  title =        "The {OpenCL} {C} Specification Version: 2.0 Document
                 Revision: 33",
  howpublished = "Web document.",
  pages =        "205",
  day =          "13",
  month =        apr,
  year =         "2016",
  bibdate =      "Mon Apr 16 14:05:49 2018",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib;
                 https://www.math.utah.edu/pub/tex/bib/pvm.bib",
  URL =          "https://www.khronos.org/registry/OpenCL/specs/opencl-2.0-openclc.pdf",
  acknowledgement = ack-nhfb,
  remark =       "Section 6.1.3.2 Math Functions, pages 74ff, defines a
                 function repertoire extended beyond that of ISO C,
                 including {\tt acospi}, {\tt asinpi}, {\tt atanpi},
                 {\tt atan2pi}, {\tt cospi}, {\tt sinpi}, {\tt tanpi},
                 {\tt cospi}, {\tt fract}, {\tt lgamma\_r}, {\tt mad}
                 (approximation to {\tt a * b + c}), {\tt minmag}, {\tt
                 pown}, {\tt rootn}, {\tt sincos}, {\tt sinpi}, and {\tt
                 tanpi}.",
}


bibtype: inproceedings
filename: elefunt.bib
label: Navas-Palencia:2016:CCH
author: Guillermo Navas-Palencia and Argimiro Arratia
crossref: Greuel:2016:MSI
pages: 241--248
year: 2016
DOI: https://doi.org/10.1007/978-3-319-42432-3_30
bibdate: Mon Feb 5 08:27:34 2018
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2018.02.05 08:27:34 ???
acknowledgement: ack-nhfb
rawauthor: Guillermo Navas-Palencia and Argimiro Arratia
title:
On the Computation of Confluent Hypergeometric Functions for Large Imaginary Part of Parameters $b$ and $z$

rawtitle:
On the Computation of Confluent Hypergeometric Functions for Large Imaginary Part of Parameters $b$ and $z$

entry:
@InProceedings{Navas-Palencia:2016:CCH,
  author =       "Guillermo Navas-Palencia and Argimiro Arratia",
  title =        "On the Computation of Confluent Hypergeometric
                 Functions for Large Imaginary Part of Parameters $b$
                 and $z$",
  crossref =     "Greuel:2016:MSI",
  pages =        "241--248",
  year =         "2016",
  DOI =          "https://doi.org/10.1007/978-3-319-42432-3_30",
  bibdate =      "Mon Feb 5 08:27:34 2018",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: elefunt.bib
label: OSullivan:2016:ZD
author: Cormac O'Sullivan
title: Zeros of the dilogarithm
journal: j-MATH-COMPUT
volume: 85
number: 302
pages: 2967--2993
month: nov
monthnumber: 11
year: 2016
CODEN: MCMPAF
DOI: https://doi.org/10.1090/mcom/3065
ISSN: 0025-5718 (print), 1088-6842 (electronic) OR 00255718 (print), 10886842 (electronic)
ISSNL: 0025-5718 OR 00255718
bibdate: Sat Nov 5 12:22:19 MDT 2016
bibtimestamp: 2016.11.05 12:22:19 MDT
fjournal: Mathematics of Computation
acknowledgement: ack-nhfb
journalURL: http://www.ams.org/mcom/
rawauthor: Cormac O'Sullivan
rawtitle: Zeros of the dilogarithm
bibsource:
http://www.ams.org/mcom/2016-85-302; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp2010.bib

URL:
http://www.ams.org/journals/mcom/2016-85-302/S0025-5718-2016-03065-3; http://www.ams.org/journals/mcom/2016-85-302/S0025-5718-2016-03065-3/S0025-5718-2016-03065-3.pdf; http://www.ams.org/mathscinet/search/author.html?mrauthid=658848

entry:
@Article{OSullivan:2016:ZD,
  author =       "Cormac O'Sullivan",
  title =        "Zeros of the dilogarithm",
  journal =      j-MATH-COMPUT,
  volume =       "85",
  number =       "302",
  pages =        "2967--2993",
  month =        nov,
  year =         "2016",
  CODEN =        "MCMPAF",
  DOI =          "https://doi.org/10.1090/mcom/3065",
  ISSN =         "0025-5718 (print), 1088-6842 (electronic)",
  ISSN-L =       "0025-5718",
  bibdate =      "Sat Nov 5 12:22:19 MDT 2016",
  bibsource =    "http://www.ams.org/mcom/2016-85-302;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/mathcomp2010.bib",
  URL =          "http://www.ams.org/journals/mcom/2016-85-302/S0025-5718-2016-03065-3;
                 http://www.ams.org/journals/mcom/2016-85-302/S0025-5718-2016-03065-3/S0025-5718-2016-03065-3.pdf;
                 http://www.ams.org/mathscinet/search/author.html?mrauthid=658848",
  acknowledgement = ack-nhfb,
  fjournal =     "Mathematics of Computation",
  journal-URL =  "http://www.ams.org/mcom/",
}


bibtype: article
filename: elefunt.bib
label: Ozcag:2016:RPI
author: Emin Ozcag and Inci Ege
title: Remarks on polygamma and incomplete gamma type functions
journal: j-J-NUMBER-THEORY
volume: 169
pages: 369--387
month: dec
monthnumber: 12
year: 2016
CODEN: JNUTA9
DOI: https://doi.org/10.1016/j.jnt.2016.05.021
ISSN: 0022-314X (print), 1096-1658 (electronic) OR 0022314X (print), 10961658 (electronic)
ISSNL: 0022-314X OR 0022314X
bibdate: Wed Jul 15 08:49:24 MDT 2020
bibtimestamp: 2020.07.15 08:49:24 MDT
URL: http://www.sciencedirect.com/science/article/pii/S0022314X16301378
fjournal: Journal of Number Theory
acknowledgement: ack-nhfb
ajournal: J. Number Theory
journalURL: http://www.sciencedirect.com/science/journal/0022314X
rawauthor: Emin {\"O}zc{\d{a}}{\u{g}} and {\.I}nci Ege
rawtitle: Remarks on polygamma and incomplete gamma type functions
entry:
@Article{Ozcag:2016:RPI,
  author =       "Emin {\"O}zc{\d{a}}{\u{g}} and {\.I}nci Ege",
  title =        "Remarks on polygamma and incomplete gamma type
                 functions",
  journal =      j-J-NUMBER-THEORY,
  volume =       "169",
  number =       "??",
  pages =        "369--387",
  month =        dec,
  year =         "2016",
  CODEN =        "JNUTA9",
  DOI =          "https://doi.org/10.1016/j.jnt.2016.05.021",
  ISSN =         "0022-314X (print), 1096-1658 (electronic)",
  ISSN-L =       "0022-314X",
  bibdate =      "Wed Jul 15 08:49:24 MDT 2020",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jnumbertheory2010.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0022314X16301378",
  acknowledgement = ack-nhfb,
  ajournal =     "J. Number Theory",
  fjournal =     "Journal of Number Theory",
  journal-URL =  "http://www.sciencedirect.com/science/journal/0022314X",
}


bibtype: article
filename: elefunt.bib
label: Paris:2016:UAE
author: R. B. Paris
title: A uniform asymptotic expansion for the incomplete gamma functions revisited
journal: arxiv.org
pages: 1--9
day: 2
month: nov
monthnumber: 11
year: 2016
bibdate: Sat Feb 18 09:13:43 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2017.02.18 09:13:43 ???
URL: https://arxiv.org/abs/1611.00548
remark: Page 9 gives corrections to cite [Eq. 8.12.18--8.12.20]Olver:2010:NHM.
acknowledgement: ack-nhfb
rawauthor: R. B. Paris
rawtitle: A uniform asymptotic expansion for the incomplete gamma functions revisited
abstract:
A new uniform asymptotic expansion for the incomplete gamma function $ Gamma (a, z) $ valid for large values of $z$ was given by the author in cite Paris:2002:UAE. This expansion contains a complementary error function of an argument measuring transition across the point $ z = a$, with easily computable coefficients that do not involve a removable singularity in the neighbourhood of this point. In this note we correct a misprint in the listing of certain coefficients in this expansion and discuss in more detail the situation corresponding to $ Gamma (a, a)$.

rawabstract:
A new uniform asymptotic expansion for the incomplete gamma function $ \Gamma (a, z) $ valid for large values of $z$ was given by the author in \cite{Paris:2002:UAE}. This expansion contains a complementary error function of an argument measuring transition across the point $ z = a$, with easily computable coefficients that do not involve a removable singularity in the neighbourhood of this point. In this note we correct a misprint in the listing of certain coefficients in this expansion and discuss in more detail the situation corresponding to $ \Gamma (a, a)$.

entry:
@Article{Paris:2016:UAE,
  author =       "R. B. Paris",
  title =        "A uniform asymptotic expansion for the incomplete
                 gamma functions revisited",
  journal =      "arxiv.org",
  volume =       "??",
  number =       "??",
  pages =        "1--9",
  day =          "2",
  month =        nov,
  year =         "2016",
  bibdate =      "Sat Feb 18 09:13:43 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "https://arxiv.org/abs/1611.00548",
  abstract =     "A new uniform asymptotic expansion for the incomplete
                 gamma function $ \Gamma (a, z) $ valid for large values
                 of $z$ was given by the author in
                 \cite{Paris:2002:UAE}. This expansion contains a
                 complementary error function of an argument measuring
                 transition across the point $ z = a$, with easily
                 computable coefficients that do not involve a removable
                 singularity in the neighbourhood of this point. In this
                 note we correct a misprint in the listing of certain
                 coefficients in this expansion and discuss in more
                 detail the situation corresponding to $ \Gamma (a,
                 a)$.",
  acknowledgement = ack-nhfb,
  remark =       "Page 9 gives corrections to \cite[Eq.
                 8.12.18--8.12.20]{Olver:2010:NHM}.",
}


bibtype: article
filename: elefunt.bib
label: Piparo:2016:CPT
author: D. Piparo and V. Innocente
title: The CptnHook Profiler --- a tool to investigate usage patterns of mathematical functions
journal: Journal of Physics: Conference Series
volume: 762
pages: 012038:1--012038:
month: oct
monthnumber: 10
year: 2016
DOI: https://doi.org/10.1088/1742-6596/762/1/012038
ISSN: 1742-6588 (print), 1742-6596 (electronic) OR 17426588 (print), 17426596 (electronic)
ISSNL: 1742-6588 OR 17426588
bibdate: Thu Sep 19 14:53:02 2024
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2024.09.19 14:53:02 ???
acknowledgement: ack-nhfb
rawauthor: D. Piparo and V. Innocente
rawtitle: The {CptnHook Profiler} --- a tool to investigate usage patterns of mathematical functions
entry:
@Article{Piparo:2016:CPT,
  author =       "D. Piparo and V. Innocente",
  title =        "The {CptnHook Profiler} --- a tool to investigate
                 usage patterns of mathematical functions",
  journal =      "Journal of Physics: Conference Series",
  volume =       "762",
  pages =        "012038:1--012038:",
  month =        oct,
  year =         "2016",
  CODEN =        "????",
  DOI =          "https://doi.org/10.1088/1742-6596/762/1/012038",
  ISSN =         "1742-6588 (print), 1742-6596 (electronic)",
  ISSN-L =       "1742-6588",
  bibdate =      "Thu Sep 19 14:53:02 2024",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
}


