# bibtab

bibtype: article
filename: elefunt.bib
label: Gil:2007:NSS
author: Amparo Gil and Javier Segura and Nico M. Temme
title: Numerically satisfactory solutions of hypergeometric recursions
journal: j-MATH-COMPUT
volume: 76
number: 259
pages: 1449--1468
month: jul
monthnumber: 7
year: 2007
CODEN: MCMPAF
ISSN: 0025-5718 (print), 1088-6842 (electronic) OR 00255718 (print), 10886842 (electronic)
ISSNL: 0025-5718 OR 00255718
bibdate: Tue Jul 8 06:24:22 MDT 2008
bibtimestamp: 2008.07.08 06:24:22 MDT
fjournal: Mathematics of Computation
acknowledgement: ack-nhfb
journalURL: http://www.ams.org/mcom/
rawauthor: Amparo Gil and Javier Segura and Nico M. Temme
rawtitle: Numerically satisfactory solutions of hypergeometric recursions
bibsource:
http://www.ams.org/mcom/2007-76-259; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcomp2000.bib

URL:
http://www.ams.org/mcom/2007-76-259/S0025-5718-07-01918-7/home.html; http://www.ams.org/mcom/2007-76-259/S0025-5718-07-01918-7/S0025-5718-07-01918-7.dvi; http://www.ams.org/mcom/2007-76-259/S0025-5718-07-01918-7/S0025-5718-07-01918-7.pdf; http://www.ams.org/mcom/2007-76-259/S0025-5718-07-01918-7/S0025-5718-07-01918-7.ps

entry:
@Article{Gil:2007:NSS,
  author =       "Amparo Gil and Javier Segura and Nico M. Temme",
  title =        "Numerically satisfactory solutions of hypergeometric
                 recursions",
  journal =      j-MATH-COMPUT,
  volume =       "76",
  number =       "259",
  pages =        "1449--1468",
  month =        jul,
  year =         "2007",
  CODEN =        "MCMPAF",
  ISSN =         "0025-5718 (print), 1088-6842 (electronic)",
  ISSN-L =       "0025-5718",
  bibdate =      "Tue Jul 8 06:24:22 MDT 2008",
  bibsource =    "http://www.ams.org/mcom/2007-76-259;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/mathcomp2000.bib",
  URL =          "http://www.ams.org/mcom/2007-76-259/S0025-5718-07-01918-7/home.html;
                 http://www.ams.org/mcom/2007-76-259/S0025-5718-07-01918-7/S0025-5718-07-01918-7.dvi;
                 http://www.ams.org/mcom/2007-76-259/S0025-5718-07-01918-7/S0025-5718-07-01918-7.pdf;
                 http://www.ams.org/mcom/2007-76-259/S0025-5718-07-01918-7/S0025-5718-07-01918-7.ps",
  acknowledgement = ack-nhfb,
  fjournal =     "Mathematics of Computation",
  journal-URL =  "http://www.ams.org/mcom/",
}


bibtype: article
filename: elefunt.bib
label: Glaser:2007:FAC
author: Andreas Glaser and Xiangtao Liu and Vladimir Rokhlin
title: A Fast Algorithm for the Calculation of the Roots of Special Functions
journal: j-SIAM-J-SCI-COMP
volume: 29
number: 4
pages: 1420--1438
year: 2007
CODEN: SJOCE3
DOI: https://doi.org/10.1137/06067016X
ISSN: 1064-8275 (print), 1095-7197 (electronic) OR 10648275 (print), 10957197 (electronic)
ISSNL: 1064-8275 OR 10648275
bibdate: Wed May 19 10:43:53 MDT 2010
bibsource: http://epubs.siam.org/sam-bin/dbq/toc/SISC/29/4; https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2010.05.19 10:43:53 MDT
fjournal: SIAM Journal on Scientific Computing
acknowledgement: ack-nhfb
journalURL: http://epubs.siam.org/sisc
rawauthor: Andreas Glaser and Xiangtao Liu and Vladimir Rokhlin
rawtitle: A Fast Algorithm for the Calculation of the Roots of Special Functions
abstract:
We describe a procedure for the determination of the roots of functions satisfying second-order ordinary differential equations, including the classical special functions. The scheme is based on a combination of the Prufer transform with the classical Taylor series method for the solution of ordinary differential equations and requires $ O(1) $ operations for the determination of each root. When the functions in question are classical orthogonal polynomials (Legendre, Hermite, etc.), the techniques presented here also provide tools for the evaluation of the weights for the associated Gaussian quadratures. The performance of the scheme for several classical special functions (prolate spheroidal wave functions, Bessel functions, and Legendre, Hermite, and Laguerre polynomials) is illustrated with numerical examples.

rawabstract:
We describe a procedure for the determination of the roots of functions satisfying second-order ordinary differential equations, including the classical special functions. The scheme is based on a combination of the Pr{\"u}fer transform with the classical Taylor series method for the solution of ordinary differential equations and requires $ O(1) $ operations for the determination of each root. When the functions in question are classical orthogonal polynomials (Legendre, Hermite, etc.), the techniques presented here also provide tools for the evaluation of the weights for the associated Gaussian quadratures. The performance of the scheme for several classical special functions (prolate spheroidal wave functions, Bessel functions, and Legendre, Hermite, and Laguerre polynomials) is illustrated with numerical examples.

entry:
@Article{Glaser:2007:FAC,
  author =       "Andreas Glaser and Xiangtao Liu and Vladimir Rokhlin",
  title =        "A Fast Algorithm for the Calculation of the Roots of
                 Special Functions",
  journal =      j-SIAM-J-SCI-COMP,
  volume =       "29",
  number =       "4",
  pages =        "1420--1438",
  month =        "????",
  year =         "2007",
  CODEN =        "SJOCE3",
  DOI =          "https://doi.org/10.1137/06067016X",
  ISSN =         "1064-8275 (print), 1095-7197 (electronic)",
  ISSN-L =       "1064-8275",
  bibdate =      "Wed May 19 10:43:53 MDT 2010",
  bibsource =    "http://epubs.siam.org/sam-bin/dbq/toc/SISC/29/4;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  abstract =     "We describe a procedure for the determination of the
                 roots of functions satisfying second-order ordinary
                 differential equations, including the classical special
                 functions. The scheme is based on a combination of the
                 Pr{\"u}fer transform with the classical Taylor series
                 method for the solution of ordinary differential
                 equations and requires $ O(1) $ operations for the
                 determination of each root. When the functions in
                 question are classical orthogonal polynomials
                 (Legendre, Hermite, etc.), the techniques presented
                 here also provide tools for the evaluation of the
                 weights for the associated Gaussian quadratures. The
                 performance of the scheme for several classical special
                 functions (prolate spheroidal wave functions, Bessel
                 functions, and Legendre, Hermite, and Laguerre
                 polynomials) is illustrated with numerical examples.",
  acknowledgement = ack-nhfb,
  fjournal =     "SIAM Journal on Scientific Computing",
  journal-URL =  "http://epubs.siam.org/sisc",
}


bibtype: book
filename: elefunt.bib
label: Gradshteyn:2007:TIS
author: I. S. Gradshteyn and I. M. Ryzhik and Alan Jeffrey and Daniel Zwillinger
title: Table of Integrals, Series and Products
publisher: pub-ACADEMIC
address: pub-ACADEMIC:adr
edition: Seventh
pages: xlv + 1171
year: 2007
ISBN: 0-12-373637-4 (hardcover) OR 0123736374 (hardcover)
ISBN13: 978-0-12-373637-6 (hardcover) OR 9780123736376 (hardcover)
LCCN: QA55 .G6613 2007
bibdate: Thu Feb 18 12:04:10 MST 2010
bibtimestamp: 2010.02.18 12:04:10 MST
remark: Previous edition 2000. Includes CD-ROM.
subject: Mathematics; Tables
acknowledgement: ack-nhfb
rawauthor: I. S. Gradshteyn and I. M. Ryzhik and Alan Jeffrey and Daniel Zwillinger
rawtitle: Table of Integrals, Series and Products
bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/master.bib; https://www.math.utah.edu/pub/tex/bib/mathcw.bib; prodorbis.library.yale.edu:7090/voyager

TOC:
0 Introduction 1 Elementary Functions 2 Indefinite Integrals of Elementary Functions 3 Definite Integrals of Elementary Functions 4.Combinations involving trigonometric and hyperbolic functions and power 5 Indefinite Integrals of Special Functions 6 Definite Integrals of Special Functions 7.Associated Legendre Functions 8 Special Functions 9 Hypergeometric Functions 10 Vector Field Theory 11 Algebraic Inequalities 12 Integral Inequalities 13 Matrices and related results 14 Determinants 15 Norms 16 Ordinary differential equations 17 Fourier, Laplace, and Mellin Transforms 18 The z-transform

rawTOC:
0 Introduction \\ 1 Elementary Functions \\ 2 Indefinite Integrals of Elementary Functions \\ 3 Definite Integrals of Elementary Functions \\ 4.Combinations involving trigonometric and hyperbolic functions and power \\ 5 Indefinite Integrals of Special Functions \\ 6 Definite Integrals of Special Functions \\ 7.Associated Legendre Functions \\ 8 Special Functions \\ 9 Hypergeometric Functions \\ 10 Vector Field Theory \\ 11 Algebraic Inequalities \\ 12 Integral Inequalities \\ 13 Matrices and related results \\ 14 Determinants \\ 15 Norms \\ 16 Ordinary differential equations \\ 17 Fourier, Laplace, and Mellin Transforms \\ 18 The z-transform

entry:
@Book{Gradshteyn:2007:TIS,
  author =       "I. S. Gradshteyn and I. M. Ryzhik and Alan Jeffrey and
                 Daniel Zwillinger",
  title =        "Table of Integrals, Series and Products",
  publisher =    pub-ACADEMIC,
  address =      pub-ACADEMIC:adr,
  edition =      "Seventh",
  pages =        "xlv + 1171",
  year =         "2007",
  ISBN =         "0-12-373637-4 (hardcover)",
  ISBN-13 =      "978-0-12-373637-6 (hardcover)",
  LCCN =         "QA55 .G6613 2007",
  bibdate =      "Thu Feb 18 12:04:10 MST 2010",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/master.bib;
                 https://www.math.utah.edu/pub/tex/bib/mathcw.bib;
                 prodorbis.library.yale.edu:7090/voyager",
  acknowledgement = ack-nhfb,
  remark =       "Previous edition 2000. Includes CD-ROM.",
  subject =      "Mathematics; Tables",
  tableofcontents = "0 Introduction \\
                 1 Elementary Functions \\
                 2 Indefinite Integrals of Elementary Functions \\
                 3 Definite Integrals of Elementary Functions \\
                 4.Combinations involving trigonometric and hyperbolic
                 functions and power \\
                 5 Indefinite Integrals of Special Functions \\
                 6 Definite Integrals of Special Functions \\
                 7.Associated Legendre Functions \\
                 8 Special Functions \\
                 9 Hypergeometric Functions \\
                 10 Vector Field Theory \\
                 11 Algebraic Inequalities \\
                 12 Integral Inequalities \\
                 13 Matrices and related results \\
                 14 Determinants \\
                 15 Norms \\
                 16 Ordinary differential equations \\
                 17 Fourier, Laplace, and Mellin Transforms \\
                 18 The z-transform",
  xxauthor =     "I. S. (Izrail Solomonovich) Gradshteyn and I. M.
                 (Iosif Moiseevich) Ryzhik and Alan Jeffrey and Daniel
                 Zwillinger",
}


bibtype: article
filename: elefunt.bib
label: Guseinov:2007:UTE
author: I. I. Guseinov and B. A. Mamedov
title: Unified treatment for the evaluation of generalized complete and incomplete gamma functions
journal: j-J-COMPUT-APPL-MATH
volume: 202
number: 2
pages: 435--439
day: 15
month: may
monthnumber: 5
year: 2007
CODEN: JCAMDI
ISSN: 0377-0427 (print), 1879-1778 (electronic) OR 03770427 (print), 18791778 (electronic)
ISSNL: 0377-0427 OR 03770427
bibdate: Sat Feb 25 13:13:14 MST 2017
bibtimestamp: 2017.02.25 13:13:14 MST
URL: http://www.sciencedirect.com/science/article/pii/S0377042706001506
fjournal: Journal of Computational and Applied Mathematics
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03770427
rawauthor: I. I. Guseinov and B. A. Mamedov
rawtitle: Unified treatment for the evaluation of generalized complete and incomplete gamma functions
entry:
@Article{Guseinov:2007:UTE,
  author =       "I. I. Guseinov and B. A. Mamedov",
  title =        "Unified treatment for the evaluation of generalized
                 complete and incomplete gamma functions",
  journal =      j-J-COMPUT-APPL-MATH,
  volume =       "202",
  number =       "2",
  pages =        "435--439",
  day =          "15",
  month =        may,
  year =         "2007",
  CODEN =        "JCAMDI",
  ISSN =         "0377-0427 (print), 1879-1778 (electronic)",
  ISSN-L =       "0377-0427",
  bibdate =      "Sat Feb 25 13:13:14 MST 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2005.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0377042706001506",
  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: Hernandez:2007:MPO
author: M. A. Hernandez and N. Romero
title: Methods with prefixed order for approximating square roots with global and general convergence
journal: j-APPL-MATH-COMP
volume: 194
number: 2
pages: 346--353
day: 15
month: dec
monthnumber: 12
year: 2007
CODEN: AMHCBQ
ISSN: 0096-3003 (print), 1873-5649 (electronic) OR 00963003 (print), 18735649 (electronic)
ISSNL: 0096-3003 OR 00963003
bibdate: Sat Jul 12 09:03:09 MDT 2008
bibtimestamp: 2008.07.12 09:03:09 MDT
fjournal: Applied Mathematics and Computation
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00963003
rawauthor: M. A. Hern{\'a}ndez and N. Romero
rawtitle: Methods with prefixed order for approximating square roots with global and general convergence
bibsource:
http://www.sciencedirect.com/science/journal/00963003; https://www.math.utah.edu/pub/tex/bib/applmathcomput2005.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/fparith.bib

entry:
@Article{Hernandez:2007:MPO,
  author =       "M. A. Hern{\'a}ndez and N. Romero",
  title =        "Methods with prefixed order for approximating square
                 roots with global and general convergence",
  journal =      j-APPL-MATH-COMP,
  volume =       "194",
  number =       "2",
  pages =        "346--353",
  day =          "15",
  month =        dec,
  year =         "2007",
  CODEN =        "AMHCBQ",
  ISSN =         "0096-3003 (print), 1873-5649 (electronic)",
  ISSN-L =       "0096-3003",
  bibdate =      "Sat Jul 12 09:03:09 MDT 2008",
  bibsource =    "http://www.sciencedirect.com/science/journal/00963003;
                 https://www.math.utah.edu/pub/tex/bib/applmathcomput2005.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied Mathematics and Computation",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00963003",
}


bibtype: article
filename: elefunt.bib
label: Kalmykov:2007:AOEa
author: M. Y. Kalmykov and B. F. L. Ward and Y. Yost
journal: j-J-HIGH-ENERGY-PHYS
volume: 2007
number: 02
pages: 040--??
year: 2007
CODEN: JHEPAB
ISSN: 1126-6708 OR 11266708
ISSNL: 1029-8479 OR 10298479
MRclass: 33C05 (33B30)
MRnumber: MR2318011 (2009g:33004)
bibdate: Thu Dec 01 09:16:04 2011
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2011.12.01 09:16:04 ???
fjournal: Journal of High Energy Physics
acknowledgement: ack-nhfb
ajournal: JHEP
rawauthor: M. Y. Kalmykov and B. F. L. Ward and Y. Yost
title:
All order $ epsilon $-expansion of Gauss hypergeometric functions with integer and half / integer values of parameters

rawtitle:
All order $ \epsilon $-expansion of {Gauss} hypergeometric functions with integer and half\slash integer values of parameters

entry:
@Article{Kalmykov:2007:AOEa,
  author =       "M. Y. Kalmykov and B. F. L. Ward and Y. Yost",
  title =        "All order $ \epsilon $-expansion of {Gauss}
                 hypergeometric functions with integer and half\slash
                 integer values of parameters",
  journal =      j-J-HIGH-ENERGY-PHYS,
  volume =       "2007",
  number =       "02",
  pages =        "040--??",
  month =        "????",
  year =         "2007",
  CODEN =        "JHEPAB",
  ISSN =         "1126-6708",
  ISSN-L =       "1029-8479",
  MRclass =      "33C05 (33B30)",
  MRnumber =     "MR2318011 (2009g:33004)",
  bibdate =      "Thu Dec 01 09:16:04 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
  ajournal =     "JHEP",
  fjournal =     "Journal of High Energy Physics",
  pagecount =    "21",
}


bibtype: article
filename: elefunt.bib
label: Kalmykov:2007:AOEb
author: M. Y. Kalmykov and B. F. L. Ward and Y. Yost
journal: j-J-HIGH-ENERGY-PHYS
volume: 2007
number: 11
pages: 009
year: 2007
CODEN: JHEPAB
ISSN: 1126-6708 OR 11266708
ISSNL: 1029-8479 OR 10298479
MRclass: 33C20 (33B30 41A58)
MRnumber: MR2362140 (2008m:33016)
bibdate: Thu Dec 01 09:16:04 2011
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2011.12.01 09:16:04 ???
fjournal: Journal of High Energy Physics
acknowledgement: ack-nhfb
ajournal: JHEP
rawauthor: M. Y. Kalmykov and B. F. L. Ward and Y. Yost
title:
On the all-order $ epsilon $-expansion of generalized hypergeometric functions with integer values of parameters

rawtitle:
On the all-order $ \epsilon $-expansion of generalized hypergeometric functions with integer values of parameters

entry:
@Article{Kalmykov:2007:AOEb,
  author =       "M. Y. Kalmykov and B. F. L. Ward and Y. Yost",
  title =        "On the all-order $ \epsilon $-expansion of generalized
                 hypergeometric functions with integer values of
                 parameters",
  journal =      j-J-HIGH-ENERGY-PHYS,
  volume =       "2007",
  number =       "11",
  pages =        "009",
  month =        "????",
  year =         "2007",
  CODEN =        "JHEPAB",
  ISSN =         "1126-6708",
  ISSN-L =       "1029-8479",
  MRclass =      "33C20 (33B30 41A58)",
  MRnumber =     "MR2362140 (2008m:33016)",
  bibdate =      "Thu Dec 01 09:16:04 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
  ajournal =     "JHEP",
  fjournal =     "Journal of High Energy Physics",
  pagecount =    "13",
}


bibtype: article
filename: elefunt.bib
label: Karagiannidis:2007:IAG
author: George Karagiannidis and Athanasios Lioumpas
title: An Improved Approximation for the Gaussian $Q$-Function
journal: j-IEEE-COMMUN-LET
volume: 11
number: 8
pages: 644--646
month: aug
monthnumber: 8
year: 2007
CODEN: ICLEF6
DOI: https://doi.org/10.1109/lcomm.2007.070470
ISSN: 1089-7798 (print), 1558-2558 (electronic) OR 10897798 (print), 15582558 (electronic)
ISSNL: 1089-7798 OR 10897798
bibdate: Sat Dec 16 16:49:58 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2017.12.16 16:49:58 ???
note: See corrections and comments cite Dyer:2008:CCI.
fjournal: IEEE Communications Letters
acknowledgement: ack-nhfb
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=4234
rawauthor: George Karagiannidis and Athanasios Lioumpas
rawnote: See corrections and comments \cite{Dyer:2008:CCI}.
rawtitle: An Improved Approximation for the {Gaussian} {$Q$}-Function
entry:
@Article{Karagiannidis:2007:IAG,
  author =       "George Karagiannidis and Athanasios Lioumpas",
  title =        "An Improved Approximation for the {Gaussian}
                 {$Q$}-Function",
  journal =      j-IEEE-COMMUN-LET,
  volume =       "11",
  number =       "8",
  pages =        "644--646",
  month =        aug,
  year =         "2007",
  CODEN =        "ICLEF6",
  DOI =          "https://doi.org/10.1109/lcomm.2007.070470",
  ISSN =         "1089-7798 (print), 1558-2558 (electronic)",
  ISSN-L =       "1089-7798",
  bibdate =      "Sat Dec 16 16:49:58 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  note =         "See corrections and comments \cite{Dyer:2008:CCI}.",
  acknowledgement = ack-nhfb,
  fjournal =     "IEEE Communications Letters",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=4234",
}


bibtype: book
filename: elefunt.bib
label: King:2007:DNC
author: Louis Vessot King
title: On the Direct Numerical Calculation of Elliptic Functions and Integrals
publisher: Mellon Press
pages: 56
year: 2007
ISBN: 1-4067-4226-0 OR 1406742260
ISBN13: 978-1-4067-4226-8 OR 9781406742268
bibdate: Wed Feb 03 08:53:04 2010
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/mathcw.bib
bibtimestamp: 2010.02.03 08:53:04 ???
URL: https://www.google.com/books/edition/On_the_Direct_Numerical_Calculation_of_E/CMM4AAAAMAAJ
acknowledgement: ack-nhfb
rawauthor: Louis Vessot King
rawtitle: On the Direct Numerical Calculation of Elliptic Functions and Integrals
note:
Dedicated to the memory of James Harkness, Peter Redpath Professor of Pure Mathematics, McGill University 1903--1923.

remark:
The AGM method for Jacobian elliptic functions was discovered by this book's author at McGill University in 1913, first published in cite King:1921:SNF, and then in a 1924 monograph, of which this is a reprint.

TOC:
Preface / v Introduction / 1 II. Historical Note on Landen's Transformation and the Various Scales of Moduli and Amplitudes / 2 III. On the Scale of Arithmetico-Geometrical Means / 6 IV. Landen's Scale of Increasing Amplitudes: $tan (phi _n + 1 - phi _n) = (b_n / a_n) tan phi _n$ / 7 (i) Calculation of $F(phi , k)$, $E(phi , k)$, $K$, and $E$ / 7 (ii) Calculation of $sn (u,k)$, $cn (u,k)$, $dn (u,k)$ in terms of the argument $u$ / 9 V. The Hyperbolic Scale of Increasing Amplitudes: $tanh (phi _n + 1 - phi _n) = (b_n / a_n) tanh (phi _n)$ / 10 (i) Calculation of $F(phi , k')$, $E(phi , k')$, etc. / 10 (ii) Calculation of $sn (u,k')$, $cn (u,k')$, $dn (u,k')$ / 11 (iii) Calculation of $sn (i u,k)$, $cn (i u,k)$, $dn (i u,k)$ / 12 VI: Gauss' Scale of Increasing Amplitudes VII: Landen's Scale of Decreasing Amplitudes: $sin (2psi _n + 1 - psi _n) = (b_n / a_n) sin psi _n$ / 14 (i) Calculation of $F(psi , k')$, $E(psi , k')$, etc. / 14 (ii) Calculation of $sn (u,k')$, $cn (u,k')$, $dn (u,k')$ / 16 VIII. The Hyperbolic Scale of Decreasing Amplitudes: $sinh (2psi _n + 1 - psi _n) = (b_n / a_n) sinh psi _n$ / 17 (i) Calculation of $F(psi , k)$, $E(psi , k)$, etc. / 17 (ii) Calculation of $sn (u,k)$, $cn (u,k)$, $dn (u,k)$ / 18 IX. On the Numerical Computation of the Third Elliptic Integral / 19 Case I. $n$ negative, between $0$ and $-k^2$ (Hyperbolic case) / 19 Case II. $n$ negative, between $-1$ and $-infty $ (Hyperbolic case) / 20 Case III. $n$ negative, between $-k^2$ and $-1$ (Circular case) / 21 First Method. (i) Use of circular recurrence formulae / 22 (ii) Use of hyperbolic recurrence formulae / 23 Second Method. (i) Use of circular recurrence formulae / 25 (ii) Use of hyperbolic recurrence formulae / 25 Case IV. $n$ positive, between $0$ and $infty $ (Circular case) / 26 First Method. (i) Use of circular recurrence formulae / 26 (ii) Use of hyperbolic recurrence formulae / 27 Second Method. (i) Use of circular recurrence formulae / 28 (ii) Use of hyperbolic recurrence formulae / 29 X. Note on the Calculation of the Third Elliptic Integral in Terms of the Complementary A.G.M. Scale / 30 Summary of Formulae / 31 Appendix, Examples 1--35 / 35

rawnote:
Dedicated to the memory of James Harkness, Peter Redpath Professor of Pure Mathematics, McGill University 1903--1923.

rawTOC:
Preface / v \\ Introduction / 1 \\ II. Historical Note on Landen's Transformation and the Various Scales of Moduli and Amplitudes / 2 \\ III. On the Scale of Arithmetico-Geometrical Means / 6 \\ IV. Landen's Scale of Increasing Amplitudes: $\tan(\phi_{n + 1} - \phi_n) = (b_n / a_n) \tan \phi_n$ / 7 \\ (i) Calculation of $F(\phi, k)$, $E(\phi, k)$, $K$, and $E$ / 7 \\ (ii) Calculation of $\sn(u,k)$, $\cn(u,k)$, $\dn(u,k)$ in terms of the argument $u$ / 9 \\ V. The Hyperbolic Scale of Increasing Amplitudes: $\tanh(\phi_{n + 1} - \phi_n) = (b_n / a_n) \tanh(\phi_n)$ / 10 \\ (i) Calculation of $F(\phi, k')$, $E(\phi, k')$, etc. / 10 \\ (ii) Calculation of $\sn(u,k')$, $\cn(u,k')$, $\dn(u,k')$ / 11 \\ (iii) Calculation of $\sn(i u,k)$, $\cn(i u,k)$, $\dn(i u,k)$ / 12 \\ VI: Gauss' Scale of Increasing Amplitudes \\ VII: Landen's Scale of Decreasing Amplitudes: $\sin(2\psi_{n + 1} - \psi_n) = (b_n / a_n) \sin \psi_n$ / 14 \\ (i) Calculation of $F(\psi, k')$, $E(\psi, k')$, etc. / 14 \\ (ii) Calculation of $\sn(u,k')$, $\cn(u,k')$, $\dn(u,k')$ / 16 \\ VIII. The Hyperbolic Scale of Decreasing Amplitudes: $\sinh(2\psi_{n + 1} - \psi_n) = (b_n / a_n) \sinh \psi_n$ / 17 \\ (i) Calculation of $F(\psi, k)$, $E(\psi, k)$, etc. / 17 \\ (ii) Calculation of $\sn(u,k)$, $\cn(u,k)$, $\dn(u,k)$ / 18 \\ IX. On the Numerical Computation of the Third Elliptic Integral / 19 \\ Case I. $n$ negative, between $0$ and $-k^2$ (Hyperbolic case) / 19 \\ Case II. $n$ negative, between $-1$ and $-\infty$ (Hyperbolic case) / 20 \\ Case III. $n$ negative, between $-k^2$ and $-1$ (Circular case) / 21 \\ First Method. (i) Use of circular recurrence formulae / 22 \\ (ii) Use of hyperbolic recurrence formulae / 23 \\ Second Method. (i) Use of circular recurrence formulae / 25 \\ (ii) Use of hyperbolic recurrence formulae / 25 \\ Case IV. $n$ positive, between $0$ and $\infty$ (Circular case) / 26 \\ First Method. (i) Use of circular recurrence formulae / 26 \\ (ii) Use of hyperbolic recurrence formulae / 27 \\ Second Method. (i) Use of circular recurrence formulae / 28 \\ (ii) Use of hyperbolic recurrence formulae / 29 \\ X. Note on the Calculation of the Third Elliptic Integral in Terms of the Complementary A.G.M. Scale / 30 \\ Summary of Formulae / 31 \\ Appendix, Examples 1--35 / 35

entry:
@Book{King:2007:DNC,
  author =       "Louis Vessot King",
  title =        "On the Direct Numerical Calculation of Elliptic
                 Functions and Integrals",
  publisher =    "Mellon Press",
  address =      "????",
  pages =        "56",
  year =         "2007",
  ISBN =         "1-4067-4226-0",
  ISBN-13 =      "978-1-4067-4226-8",
  LCCN =         "",
  bibdate =      "Wed Feb 03 08:53:04 2010",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/mathcw.bib",
  note =         "Dedicated to the memory of James Harkness, Peter
                 Redpath Professor of Pure Mathematics, McGill
                 University 1903--1923.",
  URL =          "https://www.google.com/books/edition/On_the_Direct_Numerical_Calculation_of_E/CMM4AAAAMAAJ",
  acknowledgement = ack-nhfb,
  remark =       "The AGM method for Jacobian elliptic functions was
                 discovered by this book's author at McGill University
                 in 1913, first published in \cite{King:1921:SNF}, and
                 then in a 1924 monograph, of which this is a reprint.",
  tableofcontents = "Preface / v \\
                 Introduction / 1 \\
                 II. Historical Note on Landen's Transformation and the
                 Various Scales of Moduli and Amplitudes / 2 \\
                 III. On the Scale of Arithmetico-Geometrical Means / 6
                 \\
                 IV. Landen's Scale of Increasing Amplitudes:
                 $\tan(\phi_{n + 1} - \phi_n) = (b_n / a_n) \tan \phi_n$
                 / 7 \\
                 (i) Calculation of $F(\phi, k)$, $E(\phi, k)$, $K$, and
                 $E$ / 7 \\
                 (ii) Calculation of $\sn(u,k)$, $\cn(u,k)$, $\dn(u,k)$
                 in terms of the argument $u$ / 9 \\
                 V. The Hyperbolic Scale of Increasing Amplitudes:
                 $\tanh(\phi_{n + 1} - \phi_n) = (b_n / a_n)
                 \tanh(\phi_n)$ / 10 \\
                 (i) Calculation of $F(\phi, k')$, $E(\phi, k')$, etc. /
                 10 \\
                 (ii) Calculation of $\sn(u,k')$, $\cn(u,k')$,
                 $\dn(u,k')$ / 11 \\
                 (iii) Calculation of $\sn(i u,k)$, $\cn(i u,k)$, $\dn(i
                 u,k)$ / 12 \\
                 VI: Gauss' Scale of Increasing Amplitudes \\
                 VII: Landen's Scale of Decreasing Amplitudes:
                 $\sin(2\psi_{n + 1} - \psi_n) = (b_n / a_n) \sin
                 \psi_n$ / 14 \\
                 (i) Calculation of $F(\psi, k')$, $E(\psi, k')$, etc. /
                 14 \\
                 (ii) Calculation of $\sn(u,k')$, $\cn(u,k')$,
                 $\dn(u,k')$ / 16 \\
                 VIII. The Hyperbolic Scale of Decreasing Amplitudes:
                 $\sinh(2\psi_{n + 1} - \psi_n) = (b_n / a_n) \sinh
                 \psi_n$ / 17 \\
                 (i) Calculation of $F(\psi, k)$, $E(\psi, k)$, etc. /
                 17 \\
                 (ii) Calculation of $\sn(u,k)$, $\cn(u,k)$, $\dn(u,k)$
                 / 18 \\
                 IX. On the Numerical Computation of the Third Elliptic
                 Integral / 19 \\
                 Case I. $n$ negative, between $0$ and $-k^2$
                 (Hyperbolic case) / 19 \\
                 Case II. $n$ negative, between $-1$ and $-\infty$
                 (Hyperbolic case) / 20 \\
                 Case III. $n$ negative, between $-k^2$ and $-1$
                 (Circular case) / 21 \\
                 First Method. (i) Use of circular recurrence formulae /
                 22 \\
                 (ii) Use of hyperbolic recurrence formulae / 23 \\
                 Second Method. (i) Use of circular recurrence formulae
                 / 25 \\
                 (ii) Use of hyperbolic recurrence formulae / 25 \\
                 Case IV. $n$ positive, between $0$ and $\infty$
                 (Circular case) / 26 \\
                 First Method. (i) Use of circular recurrence formulae /
                 26 \\
                 (ii) Use of hyperbolic recurrence formulae / 27 \\
                 Second Method. (i) Use of circular recurrence formulae
                 / 28 \\
                 (ii) Use of hyperbolic recurrence formulae / 29 \\
                 X. Note on the Calculation of the Third Elliptic
                 Integral in Terms of the Complementary A.G.M. Scale /
                 30 \\
                 Summary of Formulae / 31 \\
                 Appendix, Examples 1--35 / 35",
}


bibtype: article
filename: elefunt.bib
label: Kodama:2007:RA
author: Masao Kodama
title: Remark on Algorithm 644
journal: j-TOMS
volume: 33
number: 4
pages: 28:1--28:3
month: aug
monthnumber: 8
year: 2007
CODEN: ACMSCU
DOI: https://doi.org/10.1145/1268776.1268783
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
bibdate: Mon Dec 17 18:09:13 MST 2007
bibtimestamp: 2007.12.17 18:09:13 MST
note: See cite Amos:1986:APP,Amos:1990:RPP,Amos:1995:RAP.
fjournal: ACM Transactions on Mathematical Software (TOMS)
acknowledgement: ack-nhfb
articleno: 28
journalURL: https://dl.acm.org/loi/toms
rawauthor: Masao Kodama
rawnote: See \cite{Amos:1986:APP,Amos:1990:RPP,Amos:1995:RAP}.
rawtitle: Remark on {Algorithm 644}
abstract:
This remark details correction for errors in the functions which compute the modified Bessel function of the second kind and the log of the gamma function. In both cases these errors cause a loss of precision for a small range of values of the $ nu $ argument. These routines are used in the calculation of a number of other functions within the package whose accuracy is thus similarly affected.

rawabstract:
This remark details correction for errors in the functions which compute the modified Bessel function of the second kind and the log of the gamma function. In both cases these errors cause a loss of precision for a small range of values of the $ \nu $ argument. These routines are used in the calculation of a number of other functions within the package whose accuracy is thus similarly affected.

entry:
@Article{Kodama:2007:RA,
  author =       "Masao Kodama",
  title =        "Remark on {Algorithm 644}",
  journal =      j-TOMS,
  volume =       "33",
  number =       "4",
  pages =        "28:1--28:3",
  month =        aug,
  year =         "2007",
  CODEN =        "ACMSCU",
  DOI =          "https://doi.org/10.1145/1268776.1268783",
  ISSN =         "0098-3500 (print), 1557-7295 (electronic)",
  ISSN-L =       "0098-3500",
  bibdate =      "Mon Dec 17 18:09:13 MST 2007",
  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",
  note =         "See
                 \cite{Amos:1986:APP,Amos:1990:RPP,Amos:1995:RAP}.",
  abstract =     "This remark details correction for errors in the
                 functions which compute the modified Bessel function of
                 the second kind and the log of the gamma function. In
                 both cases these errors cause a loss of precision for a
                 small range of values of the $ \nu $ argument. These
                 routines are used in the calculation of a number of
                 other functions within the package whose accuracy is
                 thus similarly affected.",
  acknowledgement = ack-nhfb,
  articleno =    "28",
  fjournal =     "ACM Transactions on Mathematical Software (TOMS)",
  journal-URL =  "https://dl.acm.org/loi/toms",
}


bibtype: article
filename: elefunt.bib
label: Kuijlaars:2007:TIH
author: A. B. J. Kuijlaars and H. Stahl and W. Van Assche and F. Wielonsky
title: Type II Hermite--Pade approximation to the exponential function
journal: j-J-COMPUT-APPL-MATH
volume: 207
number: 2
pages: 227--244
day: 15
month: oct
monthnumber: 10
year: 2007
CODEN: JCAMDI
ISSN: 0377-0427 (print), 1879-1778 (electronic) OR 03770427 (print), 18791778 (electronic)
ISSNL: 0377-0427 OR 03770427
bibdate: Sat Feb 25 13:13:18 MST 2017
bibtimestamp: 2017.02.25 13:13:18 MST
URL: http://www.sciencedirect.com/science/article/pii/S0377042706005978
fjournal: Journal of Computational and Applied Mathematics
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03770427
rawauthor: A. B. J. Kuijlaars and H. Stahl and W. {Van Assche} and F. Wielonsky
rawtitle: {Type II} {Hermite--Pad{\'e}} approximation to the exponential function
entry:
@Article{Kuijlaars:2007:TIH,
  author =       "A. B. J. Kuijlaars and H. Stahl and W. {Van Assche}
                 and F. Wielonsky",
  title =        "{Type II} {Hermite--Pad{\'e}} approximation to the
                 exponential function",
  journal =      j-J-COMPUT-APPL-MATH,
  volume =       "207",
  number =       "2",
  pages =        "227--244",
  day =          "15",
  month =        oct,
  year =         "2007",
  CODEN =        "JCAMDI",
  ISSN =         "0377-0427 (print), 1879-1778 (electronic)",
  ISSN-L =       "0377-0427",
  bibdate =      "Sat Feb 25 13:13:18 MST 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2005.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0377042706005978",
  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: Kuliamin:2007:STI
author: V. V. Kuliamin
title: Standardization and testing of implementations of mathematical functions in floating point numbers
journal: j-PROG-COMP-SOFT
volume: 33
number: 3
pages: 154--173
year: 2007
CODEN: PCSODA
DOI: https://doi.org/10.1134/S036176880703005X
ISSN: 0361-7688 (print), 1608-3261 (electronic) OR 03617688 (print), 16083261 (electronic)
ISSNL: 0361-7688 OR 03617688
bibdate: Fri Aug 08 09:01:30 2008
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2008.08.08 09:01:30 ???
fjournal: Programming and Computer Software; translation of Programmirovaniye (Moscow, USSR) Plenum
keywords: floating-point function testing and verification
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11086
rawauthor: V. V. Kuliamin
rawtitle: Standardization and testing of implementations of mathematical functions in floating point numbers
abstract:
Requirements definition and test suites development for implementations of mathematical functions in floating point arithmetic in the framework of the IEEE 754 standard are considered. A method based on this standard is proposed for defining requirements for such functions. This method can be used for the standardization of implementations of such functions; this kind of standardization extends IEEE 754. A method for designing test suites for the verification of those requirements is presented. The proposed methods are based on specific properties of the representation of floating point numbers and on some features of the functions under examination.

rawabstract:
Requirements definition and test suites development for implementations of mathematical functions in floating point arithmetic in the framework of the IEEE 754 standard are considered. A method based on this standard is proposed for defining requirements for such functions. This method can be used for the standardization of implementations of such functions; this kind of standardization extends IEEE 754. A method for designing test suites for the verification of those requirements is presented. The proposed methods are based on specific properties of the representation of floating point numbers and on some features of the functions under examination.

entry:
@Article{Kuliamin:2007:STI,
  author =       "V. V. Kuliamin",
  title =        "Standardization and testing of implementations of
                 mathematical functions in floating point numbers",
  journal =      j-PROG-COMP-SOFT,
  volume =       "33",
  number =       "3",
  pages =        "154--173",
  year =         "2007",
  CODEN =        "PCSODA",
  DOI =          "https://doi.org/10.1134/S036176880703005X",
  ISSN =         "0361-7688 (print), 1608-3261 (electronic)",
  ISSN-L =       "0361-7688",
  bibdate =      "Fri Aug 08 09:01:30 2008",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  abstract =     "Requirements definition and test suites development
                 for implementations of mathematical functions in
                 floating point arithmetic in the framework of the IEEE
                 754 standard are considered. A method based on this
                 standard is proposed for defining requirements for such
                 functions. This method can be used for the
                 standardization of implementations of such functions;
                 this kind of standardization extends IEEE 754. A method
                 for designing test suites for the verification of those
                 requirements is presented. The proposed methods are
                 based on specific properties of the representation of
                 floating point numbers and on some features of the
                 functions under examination.",
  acknowledgement = ack-nhfb,
  fjournal =     "Programming and Computer Software; translation of
                 Programmirovaniye (Moscow, USSR) Plenum",
  journal-URL =  "http://link.springer.com/journal/11086",
  keywords =     "floating-point function testing and verification",
}


bibtype: techreport
filename: elefunt.bib
label: Lefevre:2007:SNP
author: Vincent Lefevre and Jean-Michel Muller
title: Some notes on the possible under / overflow of the most common elementary functions
type: Report
institution: LIP, Ecole Normale Superieure de Lyon
address: Lyon, France
pages: 7
year: 2007
bibdate: Fri May 25 16:18:32 2007
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2007.05.25 16:18:32 ???
URL: http://prunel.ccsd.cnrs.fr/ensl-00149414
keywords: elementary functions; floating-point arithmetic; overflow; underflow
acknowledgement: ack-nhfb
rawauthor: Vincent Lef{\'e}vre and Jean-Michel Muller
rawtitle: Some notes on the possible under\slash overflow of the most common elementary functions
abstract:
The purpose of this short note is not to describe when underflow or overflow must be signalled (it is quite clear that the rules are the same as for the basic arithmetic operations). We just want to show that for some of the most common functions and floating-point formats, in many cases, we can know in advance that the results will always lie in the range of the numbers that are representable by normal floating-point numbers, so that in these cases there is no need to worry about underflow or overflow. Note that when it is not the case, an implementation is still possible using a run-time test.

rawabstract:
The purpose of this short note is not to describe when underflow or overflow must be signalled (it is quite clear that the rules are the same as for the basic arithmetic operations). We just want to show that for some of the most common functions and floating-point formats, in many cases, we can know in advance that the results will always lie in the range of the numbers that are representable by normal floating-point numbers, so that in these cases there is no need to worry about underflow or overflow. Note that when it is not the case, an implementation is still possible using a run-time test.

entry:
@TechReport{Lefevre:2007:SNP,
  author =       "Vincent Lef{\'e}vre and Jean-Michel Muller",
  title =        "Some notes on the possible under\slash overflow of the
                 most common elementary functions",
  type =         "Report",
  institution =  "LIP, {\'E}cole Normale Sup{\'e}rieure de Lyon",
  address =      "Lyon, France",
  pages =        "7",
  year =         "2007",
  bibdate =      "Fri May 25 16:18:32 2007",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://prunel.ccsd.cnrs.fr/ensl-00149414",
  abstract =     "The purpose of this short note is not to describe when
                 underflow or overflow must be signalled (it is quite
                 clear that the rules are the same as for the basic
                 arithmetic operations). We just want to show that for
                 some of the most common functions and floating-point
                 formats, in many cases, we can know in advance that the
                 results will always lie in the range of the numbers
                 that are representable by normal floating-point
                 numbers, so that in these cases there is no need to
                 worry about underflow or overflow. Note that when it is
                 not the case, an implementation is still possible using
                 a run-time test.",
  acknowledgement = ack-nhfb,
  keywords =     "elementary functions; floating-point arithmetic;
                 overflow; underflow",
}


bibtype: article
filename: elefunt.bib
label: Neher:2007:CSF
author: Markus Neher
title: Complex standard functions and their implementation in the CoStLy library
journal: j-TOMS
volume: 33
number: 1
pages: 2:1--2:27
month: mar
monthnumber: 3
year: 2007
CODEN: ACMSCU
DOI: https://doi.org/10.1145/1206040.1206042
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
bibdate: Sat Apr 14 09:48:58 MDT 2007
bibtimestamp: 2007.04.14 09:48:58 MDT
fjournal: ACM Transactions on Mathematical Software (TOMS)
acknowledgement: ack-nhfb
articleno: 2
journalURL: https://dl.acm.org/loi/toms
rawauthor: Markus Neher
rawtitle: Complex standard functions and their implementation in the {CoStLy} library
abstract:
The practical calculation of range bounds for some complex standard functions is addressed in this article. The functions under consideration are root and power functions, the exponential, trigonometric and hyperbolic functions, and their inverse functions. For such a function $f$ and a given rectangular complex interval $z$, some interval $w$ is computed that contains all function values of $f$ in $z$. This is done by expressing the real and the imaginary part of $f$ as compositions of real standard functions and then estimating the ranges of these compositions. In many cases, the inclusions are optimal, such that $w$ is the smallest rectangular interval containing the range of $f$. The algorithms presented in this article have been implemented in a C++ class library called CoStLy (Complex Standard Functions License), which is distributed under the conditions of the GNU General Public License.

rawabstract:
The practical calculation of range bounds for some complex standard functions is addressed in this article. The functions under consideration are root and power functions, the exponential, trigonometric and hyperbolic functions, and their inverse functions. For such a function $f$ and a given rectangular complex interval $z$, some interval $w$ is computed that contains all function values of $f$ in $z$. This is done by expressing the real and the imaginary part of $f$ as compositions of real standard functions and then estimating the ranges of these compositions. In many cases, the inclusions are optimal, such that $w$ is the smallest rectangular interval containing the range of $f$. The algorithms presented in this article have been implemented in a C++ class library called CoStLy (Complex Standard Functions License), which is distributed under the conditions of the GNU General Public License.

entry:
@Article{Neher:2007:CSF,
  author =       "Markus Neher",
  title =        "Complex standard functions and their implementation in
                 the {CoStLy} library",
  journal =      j-TOMS,
  volume =       "33",
  number =       "1",
  pages =        "2:1--2:27",
  month =        mar,
  year =         "2007",
  CODEN =        "ACMSCU",
  DOI =          "https://doi.org/10.1145/1206040.1206042",
  ISSN =         "0098-3500 (print), 1557-7295 (electronic)",
  ISSN-L =       "0098-3500",
  bibdate =      "Sat Apr 14 09:48:58 MDT 2007",
  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 =     "The practical calculation of range bounds for some
                 complex standard functions is addressed in this
                 article. The functions under consideration are root and
                 power functions, the exponential, trigonometric and
                 hyperbolic functions, and their inverse functions. For
                 such a function $f$ and a given rectangular complex
                 interval $z$, some interval $w$ is computed that
                 contains all function values of $f$ in $z$. This is
                 done by expressing the real and the imaginary part of
                 $f$ as compositions of real standard functions and then
                 estimating the ranges of these compositions. In many
                 cases, the inclusions are optimal, such that $w$ is the
                 smallest rectangular interval containing the range of
                 $f$.

                 The algorithms presented in this article have been
                 implemented in a C++ class library called CoStLy
                 (Complex Standard Functions License), which is
                 distributed under the conditions of the GNU General
                 Public License.",
  acknowledgement = ack-nhfb,
  articleno =    "2",
  fjournal =     "ACM Transactions on Mathematical Software (TOMS)",
  journal-URL =  "https://dl.acm.org/loi/toms",
}


bibtype: book
filename: elefunt.bib
label: Press:2007:NRA
author: William H. Press and Saul A. Teukolsky and William T. Vetterling and Brian P. Flannery
title: Numerical Recipes --- The Art of Scientific Computing
publisher: pub-CAMBRIDGE
address: pub-CAMBRIDGE:adr
edition: Third
pages: xxi + 1235
year: 2007
LCCN: QA297 .N866 2007
bibdate: Wed Dec 15 10:40:52 1993
bibtimestamp: 1993.12.15 10:40:52 ???
URL: http://www.cambridge.org/numericalrecipes
subject: numerical analysis; computer programs; science; mathematics; C++ (computer program language)
acknowledgement: ack-nhfb
rawauthor: William H. Press and Saul A. Teukolsky and William T. Vetterling and Brian P. Flannery
rawtitle: Numerical Recipes --- The Art of Scientific Computing
ISBN:
0-521-88068-8 (hardcover), 0-521-88407-1 (with source code CD ROM), 0-521-70685-8 (source code CD ROM) OR 0521880688 (hardcover), 0521884071 (with source code CD ROM), 0521706858 (source code CD ROM)

ISBN13:
978-0-521-88068-8 (hardcover), 978-0-521-88407-5 (with source code CD ROM), 978-0-521-70685-8 (source code CD ROM) OR 9780521880688 (hardcover), 9780521884075 (with source code CD ROM), 9780521706858 (source code CD ROM)

bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/master.bib; https://www.math.utah.edu/pub/tex/bib/numana2000.bib

TOC:
1. Preliminaries 2. Solution of linear algebraic equations 3. Interpolation and extrapolation 4. Integration of functions 5. Evaluation of functions 6. Special functions 7. Random numbers 8. Sorting and selection 9. Root finding and nonlinear sets of equations 10. Minimization or maximization of functions 11. Eigensystems 12. Fast Fourier Transform 13. Fourier and spectral applications 14. Statistical description of data 15. Modeling of data 16. Classification and inference 17. Integration of ordinary differential equations 18. Two-point boundary value problems 19. Integral equations and inverse theory 20. Partial differential equations 21. Computational geometry 22. Less-numerical algorithms

rawTOC:
1. Preliminaries \\ 2. Solution of linear algebraic equations \\ 3. Interpolation and extrapolation \\ 4. Integration of functions \\ 5. Evaluation of functions \\ 6. Special functions \\ 7. Random numbers \\ 8. Sorting and selection \\ 9. Root finding and nonlinear sets of equations \\ 10. Minimization or maximization of functions \\ 11. Eigensystems \\ 12. Fast Fourier Transform \\ 13. Fourier and spectral applications \\ 14. Statistical description of data \\ 15. Modeling of data \\ 16. Classification and inference \\ 17. Integration of ordinary differential equations \\ 18. Two-point boundary value problems \\ 19. Integral equations and inverse theory \\ 20. Partial differential equations \\ 21. Computational geometry \\ 22. Less-numerical algorithms

entry:
@Book{Press:2007:NRA,
  author =       "William H. Press and Saul A. Teukolsky and William T.
                 Vetterling and Brian P. Flannery",
  title =        "Numerical Recipes --- The Art of Scientific
                 Computing",
  publisher =    pub-CAMBRIDGE,
  address =      pub-CAMBRIDGE:adr,
  edition =      "Third",
  pages =        "xxi + 1235",
  year =         "2007",
  ISBN =         "0-521-88068-8 (hardcover), 0-521-88407-1 (with source
                 code CD ROM), 0-521-70685-8 (source code CD ROM)",
  ISBN-13 =      "978-0-521-88068-8 (hardcover), 978-0-521-88407-5 (with
                 source code CD ROM), 978-0-521-70685-8 (source code CD
                 ROM)",
  LCCN =         "QA297 .N866 2007",
  bibdate =      "Wed Dec 15 10:40:52 1993",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/master.bib;
                 https://www.math.utah.edu/pub/tex/bib/numana2000.bib",
  URL =          "http://www.cambridge.org/numericalrecipes",
  acknowledgement = ack-nhfb,
  subject =      "numerical analysis; computer programs; science;
                 mathematics; C++ (computer program language)",
  tableofcontents = "1. Preliminaries \\
                 2. Solution of linear algebraic equations \\
                 3. Interpolation and extrapolation \\
                 4. Integration of functions \\
                 5. Evaluation of functions \\
                 6. Special functions \\
                 7. Random numbers \\
                 8. Sorting and selection \\
                 9. Root finding and nonlinear sets of equations \\
                 10. Minimization or maximization of functions \\
                 11. Eigensystems \\
                 12. Fast Fourier Transform \\
                 13. Fourier and spectral applications \\
                 14. Statistical description of data \\
                 15. Modeling of data \\
                 16. Classification and inference \\
                 17. Integration of ordinary differential equations \\
                 18. Two-point boundary value problems \\
                 19. Integral equations and inverse theory \\
                 20. Partial differential equations \\
                 21. Computational geometry \\
                 22. Less-numerical algorithms",
}


bibtype: article
filename: elefunt.bib
label: Ren:2007:CFA
author: C. Ren and A. R. MacKenzie
journal: j-ATMOS-SCI-LETT
volume: 8
number: 3
pages: 70--73
year: 2007
DOI: https://doi.org/10.1002/asl.154
ISSN: 1530-261X OR 1530261X
ISSNL: 1530-261X OR 1530261X
bibdate: Sat Dec 16 17:25:42 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2017.12.16 17:25:42 ???
URL: http://onlinelibrary.wiley.com/doi/10.1002/asl.154/full
fjournal: Atmospheric Science Letters
acknowledgement: ack-nhfb
rawauthor: C. Ren and A. R. MacKenzie
title:
Closed-form approximations to the error and complementary error functions and their applications in atmospheric science

journalURL:
http://www.sciencedirect.com/science/journal/1530261X; http://rmets.onlinelibrary.wiley.com/hub/journal/10.1002/(ISSN)1530-261X/

rawtitle:
Closed-form approximations to the error and complementary error functions and their applications in atmospheric science

entry:
@Article{Ren:2007:CFA,
  author =       "C. Ren and A. R. MacKenzie",
  title =        "Closed-form approximations to the error and
                 complementary error functions and their applications in
                 atmospheric science",
  journal =      j-ATMOS-SCI-LETT,
  volume =       "8",
  number =       "3",
  pages =        "70--73",
  month =        "????",
  year =         "2007",
  DOI =          "https://doi.org/10.1002/asl.154",
  ISSN =         "1530-261X",
  ISSN-L =       "1530-261X",
  bibdate =      "Sat Dec 16 17:25:42 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://onlinelibrary.wiley.com/doi/10.1002/asl.154/full",
  acknowledgement = ack-nhfb,
  fjournal =     "Atmospheric Science Letters",
  journal-URL =  "http://www.sciencedirect.com/science/journal/1530261X;
                 http://rmets.onlinelibrary.wiley.com/hub/journal/10.1002/(ISSN)1530-261X/",
}


bibtype: article
filename: elefunt.bib
label: Rokhlin:2007:AFC
author: Vladimir Rokhlin and Hong Xiao
journal: j-APPL-COMPUT-HARMON-ANAL
volume: 22
number: 1
pages: 105--123
month: jan
monthnumber: 1
year: 2007
DOI: https://doi.org/10.1016/j.acha.2006.05.004
ISSN: 1063-5203 (print), 1096-603x (electronic) OR 10635203 (print), 1096603x (electronic)
ISSNL: 1063-5203 OR 10635203
bibdate: Sun Oct 31 10:00:51 2010
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2010.10.31 10:00:51 ???
keywords: approximation; asymptotic; band-limit; prolate spheroidal wave functions
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/10635203
rawauthor: Vladimir Rokhlin and Hong Xiao
title:
Approximate formulae for certain prolate spheroidal wave functions valid for large values of both order and band-limit

abstract:
We construct asymptotic formulae for the approximation of certain prolate spheroidal wave functions and of the corresponding eigenvalues. We investigate two regimes: when the ratio $ c / m $ decays, and when both $c$ and $m$ grow, but the ratio $ c / m $ stays bounded. Both the regions of validity and the accuracies of the obtained expansions are illustrated with numerical examples.

fjournal:
Applied and Computational Harmonic Analysis. Time-Frequency and Time-Scale Analysis, Wavelets, Numerical Algorithms, and Applications

rawabstract:
We construct asymptotic formulae for the approximation of certain prolate spheroidal wave functions and of the corresponding eigenvalues. We investigate two regimes: when the ratio $ c / m $ decays, and when both $c$ and $m$ grow, but the ratio $ c / m $ stays bounded. Both the regions of validity and the accuracies of the obtained expansions are illustrated with numerical examples.

rawtitle:
Approximate formulae for certain prolate spheroidal wave functions valid for large values of both order and band-limit

entry:
@Article{Rokhlin:2007:AFC,
  author =       "Vladimir Rokhlin and Hong Xiao",
  title =        "Approximate formulae for certain prolate spheroidal
                 wave functions valid for large values of both order and
                 band-limit",
  journal =      j-APPL-COMPUT-HARMON-ANAL,
  volume =       "22",
  number =       "1",
  pages =        "105--123",
  month =        jan,
  year =         "2007",
  DOI =          "https://doi.org/10.1016/j.acha.2006.05.004",
  ISSN =         "1063-5203 (print), 1096-603x (electronic)",
  ISSN-L =       "1063-5203",
  bibdate =      "Sun Oct 31 10:00:51 2010",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  abstract =     "We construct asymptotic formulae for the approximation
                 of certain prolate spheroidal wave functions and of the
                 corresponding eigenvalues. We investigate two regimes:
                 when the ratio $ c / m $ decays, and when both $c$ and
                 $m$ grow, but the ratio $ c / m $ stays bounded. Both
                 the regions of validity and the accuracies of the
                 obtained expansions are illustrated with numerical
                 examples.",
  acknowledgement = ack-nhfb,
  fjournal =     "Applied and Computational Harmonic Analysis.
                 Time-Frequency and Time-Scale Analysis, Wavelets,
                 Numerical Algorithms, and Applications",
  journal-URL =  "http://www.sciencedirect.com/science/journal/10635203",
  keywords =     "approximation; asymptotic; band-limit; prolate
                 spheroidal wave functions",
}


bibtype: article
filename: elefunt.bib
label: Schmelzer:2007:CGF
author: Thomas Schmelzer and Lloyd N. Trefethen
title: Computing the Gamma Function Using Contour Integrals and Rational Approximations
journal: j-SIAM-J-NUMER-ANAL
volume: 45
number: 2
pages: 558--571
year: 2007
CODEN: SJNAAM
DOI: https://doi.org/10.1137/050646342
ISSN: 0036-1429 (print), 1095-7170 (electronic) OR 00361429 (print), 10957170 (electronic)
ISSNL: 0036-1429 OR 00361429
bibdate: Mon Nov 24 18:03:07 MST 2008
bibtimestamp: 2008.11.24 18:03:07 MST
fjournal: SIAM Journal on Numerical Analysis
acknowledgement: ack-nhfb
journalURL: http://epubs.siam.org/sinum
rawauthor: Thomas Schmelzer and Lloyd N. Trefethen
rawtitle: Computing the Gamma Function Using Contour Integrals and Rational Approximations
bibsource:
http://siamdl.aip.org/dbt/dbt.jsp?KEY=SJNAAM&Volume=45&Issue=2; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

entry:
@Article{Schmelzer:2007:CGF,
  author =       "Thomas Schmelzer and Lloyd N. Trefethen",
  title =        "Computing the Gamma Function Using Contour Integrals
                 and Rational Approximations",
  journal =      j-SIAM-J-NUMER-ANAL,
  volume =       "45",
  number =       "2",
  pages =        "558--571",
  month =        "????",
  year =         "2007",
  CODEN =        "SJNAAM",
  DOI =          "https://doi.org/10.1137/050646342",
  ISSN =         "0036-1429 (print), 1095-7170 (electronic)",
  ISSN-L =       "0036-1429",
  bibdate =      "Mon Nov 24 18:03:07 MST 2008",
  bibsource =    "http://siamdl.aip.org/dbt/dbt.jsp?KEY=SJNAAM&Volume=45&Issue=2;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "SIAM Journal on Numerical Analysis",
  journal-URL =  "http://epubs.siam.org/sinum",
}


bibtype: book
filename: elefunt.bib
label: Slavjanov:2007:SFU
author: Sergej J. Slavjanov and Wolfgang Lay
title: Special Functions: a Unified Theory Based on Singularities
publisher: pub-OXFORD
address: pub-OXFORD:adr
pages: xvi + 293
year: 2007
ISBN: 0-19-850573-6 OR 0198505736
ISBN13: 978-0-19-850573-0 OR 9780198505730
bibdate: Tue Dec 5 11:27:46 MST 2023
bibsource: fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2023.12.05 11:27:46 MST
series: Oxford mathematical monographs; Oxford science publications
subject: Functions, Special; Fonctions speciales; Functions, Special
acknowledgement: ack-nhfb
rawauthor: Sergej J. Slavjanov and Wolfgang Lay
rawtitle: Special Functions: a Unified Theory Based on Singularities
TOC:
1: Linear second-order ODEs with polynomial coefficients Regular singularities and Fuchsian equations Regular and Fuchsian singularities Fuchsian equations and their transformations Characteristic exponents Frobenius solutions Irregular singularities and confluent equations The $s$-rank of a singularity Confluent and reduced confluent equations The $s$-homotopic transformation Asymptotic solutions at irregular singularities Canonical forms A generalization of Fuchs's theorem Confluence and reduction processes Strong and weak confluence. A confluence theorem A confluence principle Reduction of an equation Classes and types of equations Standard forms of equations Invariants of $s$-homotopic transformations Types of solutions Eigenfunctions of singular Sturm--Liouville problems Central and lateral connection problems Stokes lines at singularities. Stokes matrices Generalized Riemann scheme Applications Central two-point connection problems (CTCPs) Two regular singularities as relevant endpoints One regular singularity and one irregular singularity as the endpoints A proof Two irregular singularities 2: The hypergeometric class of equations Classification scheme General presentation Hypergeometric equation Confluent equations Reduced confluent equations Difference equations General consideration Difference equations for hypergeometric functions Confluent hypergeometric functions ldots 

shortTOC:
1: Linear second-order ODEs with polynomial coefficients 2: The hypergeometric class of equations 3: The Heun class of equations 4: Application to physical sciences 5: The Painleve class of equations Appendix A: The gamma function and related functions Appendix B: CTCPs Heun equations in general form Appendix C: Multipole Matrix elements Appendix D: SFTools Database of the special functions

rawshortTOC:
1: Linear second-order ODEs with polynomial coefficients \\ 2: The hypergeometric class of equations \\ 3: The Heun class of equations \\ 4: Application to physical sciences \\ 5: The Painlev{\'e} class of equations \\ Appendix A: The gamma function and related functions \\ Appendix B: CTCPs Heun equations in general form \\ Appendix C: Multipole Matrix elements \\ Appendix D: SFTools \\ Database of the special functions

rawTOC:
1: Linear second-order ODEs with polynomial coefficients \\ Regular singularities and Fuchsian equations \\ Regular and Fuchsian singularities \\ Fuchsian equations and their transformations \\ Characteristic exponents \\ Frobenius solutions \\ Irregular singularities and confluent equations \\ The $s$-rank of a singularity \\ Confluent and reduced confluent equations \\ The $s$-homotopic transformation \\ Asymptotic solutions at irregular singularities \\ Canonical forms \\ A generalization of Fuchs's theorem \\ Confluence and reduction processes \\ Strong and weak confluence. A confluence theorem \\ A confluence principle \\ Reduction of an equation \\ Classes and types of equations \\ Standard forms of equations \\ Invariants of $s$-homotopic transformations \\ Types of solutions \\ Eigenfunctions of singular Sturm--Liouville problems \\ Central and lateral connection problems \\ Stokes lines at singularities. Stokes matrices \\ Generalized Riemann scheme \\ Applications \\ Central two-point connection problems (CTCPs) \\ Two regular singularities as relevant endpoints \\ One regular singularity and one irregular singularity as the endpoints \\ A proof \\ Two irregular singularities \\ 2: The hypergeometric class of equations \\ Classification scheme \\ General presentation \\ Hypergeometric equation \\ Confluent equations \\ Reduced confluent equations \\ Difference equations \\ General consideration \\ Difference equations for hypergeometric functions \\ Confluent hypergeometric functions \\ \ldots{}

entry:
@Book{Slavjanov:2007:SFU,
  author =       "Sergej J. Slavjanov and Wolfgang Lay",
  title =        "Special Functions: a Unified Theory Based on
                 Singularities",
  publisher =    pub-OXFORD,
  address =      pub-OXFORD:adr,
  pages =        "xvi + 293",
  year =         "2007",
  ISBN =         "0-19-850573-6",
  ISBN-13 =      "978-0-19-850573-0",
  LCCN =         "????",
  bibdate =      "Tue Dec 5 11:27:46 MST 2023",
  bibsource =    "fsz3950.oclc.org:210/WorldCat;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  series =       "Oxford mathematical monographs; Oxford science
                 publications",
  acknowledgement = ack-nhfb,
  shorttableofcontents = "1: Linear second-order ODEs with polynomial
                 coefficients \\
                 2: The hypergeometric class of equations \\
                 3: The Heun class of equations \\
                 4: Application to physical sciences \\
                 5: The Painlev{\'e} class of equations \\
                 Appendix A: The gamma function and related functions
                 \\
                 Appendix B: CTCPs Heun equations in general form \\
                 Appendix C: Multipole Matrix elements \\
                 Appendix D: SFTools \\
                 Database of the special functions",
  subject =      "Functions, Special; Fonctions sp{\'e}ciales;
                 Functions, Special",
  tableofcontents = "1: Linear second-order ODEs with polynomial
                 coefficients \\
                 Regular singularities and Fuchsian equations \\
                 Regular and Fuchsian singularities \\
                 Fuchsian equations and their transformations \\
                 Characteristic exponents \\
                 Frobenius solutions \\
                 Irregular singularities and confluent equations \\
                 The $s$-rank of a singularity \\
                 Confluent and reduced confluent equations \\
                 The $s$-homotopic transformation \\
                 Asymptotic solutions at irregular singularities \\
                 Canonical forms \\
                 A generalization of Fuchs's theorem \\
                 Confluence and reduction processes \\
                 Strong and weak confluence. A confluence theorem \\
                 A confluence principle \\
                 Reduction of an equation \\
                 Classes and types of equations \\
                 Standard forms of equations \\
                 Invariants of $s$-homotopic transformations \\
                 Types of solutions \\
                 Eigenfunctions of singular Sturm--Liouville problems
                 \\
                 Central and lateral connection problems \\
                 Stokes lines at singularities. Stokes matrices \\
                 Generalized Riemann scheme \\
                 Applications \\
                 Central two-point connection problems (CTCPs) \\
                 Two regular singularities as relevant endpoints \\
                 One regular singularity and one irregular singularity
                 as the endpoints \\
                 A proof \\
                 Two irregular singularities \\
                 2: The hypergeometric class of equations \\
                 Classification scheme \\
                 General presentation \\
                 Hypergeometric equation \\
                 Confluent equations \\
                 Reduced confluent equations \\
                 Difference equations \\
                 General consideration \\
                 Difference equations for hypergeometric functions \\
                 Confluent hypergeometric functions \\
                 \ldots{}",
}


bibtype: article
filename: elefunt.bib
label: Srinivasan:2007:GFE
author: Gopala Krishna Srinivasan
title: The Gamma Function: An Eclectic Tour
journal: j-AMER-MATH-MONTHLY
volume: 114
number: 4
pages: 297--315
month: apr
monthnumber: 4
year: 2007
CODEN: AMMYAE
ISSN: 0002-9890 (print), 1930-0972 (electronic) OR 00029890 (print), 19300972 (electronic)
ISSNL: 0002-9890 OR 00029890
bibdate: Mon Jan 30 12:00:28 MST 2012
bibtimestamp: 2012.01.30 12:00:28 MST
URL: http://www.jstor.org/stable/27642193
fjournal: American Mathematical Monthly
acknowledgement: ack-nhfb
journalURL: https://www.jstor.org/journals/00029890.htm
rawauthor: Gopala Krishna Srinivasan
rawtitle: The Gamma Function: An Eclectic Tour
bibsource:
http://www.jstor.org/journals/00029890.html; http://www.jstor.org/stable/i27642189; https://www.math.utah.edu/pub/tex/bib/amermathmonthly2000.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

entry:
@Article{Srinivasan:2007:GFE,
  author =       "Gopala Krishna Srinivasan",
  title =        "The Gamma Function: An Eclectic Tour",
  journal =      j-AMER-MATH-MONTHLY,
  volume =       "114",
  number =       "4",
  pages =        "297--315",
  month =        apr,
  year =         "2007",
  CODEN =        "AMMYAE",
  ISSN =         "0002-9890 (print), 1930-0972 (electronic)",
  ISSN-L =       "0002-9890",
  bibdate =      "Mon Jan 30 12:00:28 MST 2012",
  bibsource =    "http://www.jstor.org/journals/00029890.html;
                 http://www.jstor.org/stable/i27642189;
                 https://www.math.utah.edu/pub/tex/bib/amermathmonthly2000.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.jstor.org/stable/27642193",
  acknowledgement = ack-nhfb,
  fjournal =     "American Mathematical Monthly",
  journal-URL =  "https://www.jstor.org/journals/00029890.htm",
}


bibtype: article
filename: elefunt.bib
label: Temme:2007:NAS
author: Nico M. Temme
title: Numerical aspects of special functions
journal: j-ACTA-NUMERICA
volume: 16
pages: 379--478
year: 2007
CODEN: ANUMFU
DOI: https://doi.org/10.1017/S0962492906330012
ISBN: 0-521-87743-1 OR 0521877431
ISBN13: 978-0-521-87743-5 OR 9780521877435
ISSN: 0962-4929 (print), 1474-0508 (electronic) OR 09624929 (print), 14740508 (electronic)
ISSNL: 0962-4929 OR 09624929
MRclass: 33F05 (65D20)
MRnumber: 2417932 (2009g:33027)
MRreviewer: Amparo Gil
bibdate: Sat Sep 24 11:37:18 MDT 2011
bibtimestamp: 2011.09.24 11:37:18 MDT
fjournal: Acta Numerica
acknowledgement: ack-nhfb
ajournal: Acta Numer.
journalURL: http://journals.cambridge.org/action/displayJournal?jid=ANU
onlinedate: 24 April 2007
rawauthor: Nico M. Temme
rawtitle: Numerical aspects of special functions
bibsource:
https://www.math.utah.edu/pub/tex/bib/actanumerica.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

abstract:
This paper describes methods that are important for the numerical evaluation of certain functions that frequently occur in applied mathematics, physics and mathematical statistics. This includes what we consider to be the basic methods, such as recurrence relations, series expansions (both convergent and asymptotic), and numerical quadrature. Several other methods are available and some of these will be discussed in less detail. Examples will be given on the use of special functions in certain problems from mathematical physics and mathematical statistics (integrals and series with special functions).

rawabstract:
This paper describes methods that are important for the numerical evaluation of certain functions that frequently occur in applied mathematics, physics and mathematical statistics. This includes what we consider to be the basic methods, such as recurrence relations, series expansions (both convergent and asymptotic), and numerical quadrature. Several other methods are available and some of these will be discussed in less detail. Examples will be given on the use of special functions in certain problems from mathematical physics and mathematical statistics (integrals and series with special functions).

entry:
@Article{Temme:2007:NAS,
  author =       "Nico M. Temme",
  title =        "Numerical aspects of special functions",
  journal =      j-ACTA-NUMERICA,
  volume =       "16",
  pages =        "379--478",
  year =         "2007",
  CODEN =        "ANUMFU",
  DOI =          "https://doi.org/10.1017/S0962492906330012",
  ISBN =         "0-521-87743-1",
  ISBN-13 =      "978-0-521-87743-5",
  ISSN =         "0962-4929 (print), 1474-0508 (electronic)",
  ISSN-L =       "0962-4929",
  MRclass =      "33F05 (65D20)",
  MRnumber =     "2417932 (2009g:33027)",
  MRreviewer =   "Amparo Gil",
  bibdate =      "Sat Sep 24 11:37:18 MDT 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/actanumerica.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  abstract =     "This paper describes methods that are important for
                 the numerical evaluation of certain functions that
                 frequently occur in applied mathematics, physics and
                 mathematical statistics. This includes what we consider
                 to be the basic methods, such as recurrence relations,
                 series expansions (both convergent and asymptotic), and
                 numerical quadrature. Several other methods are
                 available and some of these will be discussed in less
                 detail. Examples will be given on the use of special
                 functions in certain problems from mathematical physics
                 and mathematical statistics (integrals and series with
                 special functions).",
  acknowledgement = ack-nhfb,
  ajournal =     "Acta Numer.",
  fjournal =     "Acta Numerica",
  journal-URL =  "http://journals.cambridge.org/action/displayJournal?jid=ANU",
  onlinedate =   "24 April 2007",
}


bibtype: incollection
filename: elefunt.bib
label: Weniger:2007:AAT
author: Ernst Joachim Weniger
title: Asymptotic approximations to truncation errors of series representations for special functions
crossref: Iske:2007:AAP
pages: 331--348
year: 2007
MRclass: 33F05
MRnumber: MR2335174 (2008h:33051)
bibdate: Thu Dec 01 09:38:02 2011
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2011.12.01 09:38:02 ???
remark: Available as math.CA/0511074.
acknowledgement: ack-nhfb
rawauthor: Ernst Joachim Weniger
rawtitle: Asymptotic approximations to truncation errors of series representations for special functions
keywords:
Bernoulli numbers; Euler--MacLaurin formula; exponential integer $E_1(z)$; Gaussian hypergeometric series $_2F_1(a / b / c / z)$; Riemann zeta function

entry:
@InCollection{Weniger:2007:AAT,
  author =       "Ernst Joachim Weniger",
  title =        "Asymptotic approximations to truncation errors of
                 series representations for special functions",
  crossref =     "Iske:2007:AAP",
  pages =        "331--348",
  year =         "2007",
  MRclass =      "33F05",
  MRnumber =     "MR2335174 (2008h:33051)",
  bibdate =      "Thu Dec 01 09:38:02 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
  keywords =     "Bernoulli numbers; Euler--MacLaurin formula;
                 exponential integer $E_1(z)$; Gaussian hypergeometric
                 series $_2F_1(a / b / c / z)$; Riemann zeta function",
  remark =       "Available as math.CA/0511074.",
}


bibtype: book
filename: elefunt.bib
label: Agarwal:2008:OPD
author: Ravi P. Agarwal and Donal O'Regan
publisher: pub-SV
address: pub-SV:adr
pages: xiv + 410
year: 2008
ISBN: 0-387-79145-0 (paperback) OR 0387791450 (paperback)
ISBN13: 978-0-387-79145-6 (paperback) OR 9780387791456 (paperback)
bibdate: Sat Oct 30 17:22:04 MDT 2010
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib; prodorbis.library.yale.edu:7090/voyager
bibtimestamp: 2010.10.30 17:22:04 MDT
series: Universitext
subject: differential equations; differential equations, partial; Fourier analysis; boundary value problems
acknowledgement: ack-nhfb
rawauthor: Ravi P. Agarwal and Donal O'Regan
title:
Ordinary and Partial Differential Equations: with Special Functions, Fourier Series, and Boundary Value Problems

TOC:
Preface vii l. Solvable Differential Equations / 1 2. Second-Order Differential Equations / 8 3. Preliminaries to Series Solutions / 15 4. Solution at an Ordinary Point / 23 5. Solution at a Singular Point / 31 6. Solution at a Singular Point (Cont'd.) / 37 7. Legendre Polynomials and Functions / 47 8. Chebyshev, Hermite and Laguerre Polynomials / 57 9. Bessel Functions / 64 10. Hypergeometrie Functions / 75 11. Piecewise Continuous and Periodic Functions / 83 12. Orthogonal Functions and Polynomials / 90 13. Orthogonal Functions and Polynomials (Cont'd.) / 95 14. Boundary Value Problems / 104 15. Boundary Value Problems (Cont'd.) / 109 16. Green's Functions / 119 17. Regular Perturbations / 129 18. Singular Perturbations / 138 19. Sturm--Liouville Problems / 145 20. Eigenfunction Expansions / 157 21. Eigenfunction Expansions (Cont'd.) / 163 22. Convergence of the Fourier Series / 171 23. Convergence of the Fourier Series (Cont'd.) / 176 24. Fourier Series Solutions of Ordinary Differential Equations / 187 25. Partial Differential Equations / 194 26. First-Order Partial Differential Equations / 202 27. Solvable Partial Differential Equations / 210 28. The Canonical Forms / 219 29. The Method of Separation of Variables / 227 30. The One-Dimensional Heat Equation / 234 31. The One-Dimensional Heat Equation (Cont'd.) / 241 32. The One-Dimensional Wave Equation / 249 33. The One-Dimensional Wave Equation (Cont'd.) / 256 34. Laplace Equation in Two Dimensions / 266 35. Laplace Equation in Polar Coordinates / 275 36. Two-Dimensional Heat Equation / 284 37. Two-Dimensional Wave Equation / 292 38. Laplace Equation in Three Dimensions / 300 39. Laplace Equation in Three Dimensions (Cont'd.) / 306 40. Nonhomogeneous Equations / 316 41. Fourier Integral and Transforms / 323 42. Fourier Integral and Transforms (Cont'd.) / 330 43. Fourier Transform Method for Partial DEs / 338 44. Fourier Transform Method for Partial DEs (Cont'd.) / 344 45. Laplace Transforms / 354 46. Laplace Transforms (Cont'd.) / 361 47. Laplace Transform Method for Ordinary DEs / 374 48. Laplace Transform Method for Partial DEs / 384 49. Well-Posed Problems / 394 50. Verification of Solutions / 399 References for Further Reading / 405 Index / 407

rawtitle:
Ordinary and Partial Differential Equations: with Special Functions, {Fourier} Series, and Boundary Value Problems

rawTOC:
Preface vii l. Solvable Differential Equations / 1 \\ 2. Second-Order Differential Equations / 8 \\ 3. Preliminaries to Series Solutions / 15 \\ 4. Solution at an Ordinary Point / 23 \\ 5. Solution at a Singular Point / 31 \\ 6. Solution at a Singular Point (Cont'd.) / 37 \\ 7. Legendre Polynomials and Functions / 47 \\ 8. Chebyshev, Hermite and Laguerre Polynomials / 57 \\ 9. Bessel Functions / 64 \\ 10. Hypergeometrie Functions / 75 \\ 11. Piecewise Continuous and Periodic Functions / 83 \\ 12. Orthogonal Functions and Polynomials / 90 \\ 13. Orthogonal Functions and Polynomials (Cont'd.) / 95 \\ 14. Boundary Value Problems / 104 \\ 15. Boundary Value Problems (Cont'd.) / 109 \\ 16. Green's Functions / 119 \\ 17. Regular Perturbations / 129 \\ 18. Singular Perturbations / 138 \\ 19. Sturm--Liouville Problems / 145 \\ 20. Eigenfunction Expansions / 157 \\ 21. Eigenfunction Expansions (Cont'd.) / 163 \\ 22. Convergence of the Fourier Series / 171 \\ 23. Convergence of the Fourier Series (Cont'd.) / 176 \\ 24. Fourier Series Solutions of Ordinary Differential Equations / 187 \\ 25. Partial Differential Equations / 194 \\ 26. First-Order Partial Differential Equations / 202 \\ 27. Solvable Partial Differential Equations / 210 \\ 28. The Canonical Forms / 219 \\ 29. The Method of Separation of Variables / 227 \\ 30. The One-Dimensional Heat Equation / 234 \\ 31. The One-Dimensional Heat Equation (Cont'd.) / 241 \\ 32. The One-Dimensional Wave Equation / 249 \\ 33. The One-Dimensional Wave Equation (Cont'd.) / 256 \\ 34. Laplace Equation in Two Dimensions / 266 \\ 35. Laplace Equation in Polar Coordinates / 275 \\ 36. Two-Dimensional Heat Equation / 284 \\ 37. Two-Dimensional Wave Equation / 292 \\ 38. Laplace Equation in Three Dimensions / 300 \\ 39. Laplace Equation in Three Dimensions (Cont'd.) / 306 \\ 40. Nonhomogeneous Equations / 316 \\ 41. Fourier Integral and Transforms / 323 \\ 42. Fourier Integral and Transforms (Cont'd.) / 330 \\ 43. Fourier Transform Method for Partial DEs / 338 \\ 44. Fourier Transform Method for Partial DEs (Cont'd.) / 344 \\ 45. Laplace Transforms / 354 \\ 46. Laplace Transforms (Cont'd.) / 361 \\ 47. Laplace Transform Method for Ordinary DEs / 374 \\ 48. Laplace Transform Method for Partial DEs / 384 \\ 49. Well-Posed Problems / 394 \\ 50. Verification of Solutions / 399 \\ References for Further Reading / 405 \\ Index / 407

entry:
@Book{Agarwal:2008:OPD,
  author =       "Ravi P. Agarwal and Donal O'Regan",
  title =        "Ordinary and Partial Differential Equations: with
                 Special Functions, {Fourier} Series, and Boundary Value
                 Problems",
  publisher =    pub-SV,
  address =      pub-SV:adr,
  pages =        "xiv + 410",
  year =         "2008",
  ISBN =         "0-387-79145-0 (paperback)",
  ISBN-13 =      "978-0-387-79145-6 (paperback)",
  LCCN =         "????",
  bibdate =      "Sat Oct 30 17:22:04 MDT 2010",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 prodorbis.library.yale.edu:7090/voyager",
  series =       "Universitext",
  acknowledgement = ack-nhfb,
  subject =      "differential equations; differential equations,
                 partial; Fourier analysis; boundary value problems",
  tableofcontents = "Preface vii l. Solvable Differential Equations / 1
                 \\
                 2. Second-Order Differential Equations / 8 \\
                 3. Preliminaries to Series Solutions / 15 \\
                 4. Solution at an Ordinary Point / 23 \\
                 5. Solution at a Singular Point / 31 \\
                 6. Solution at a Singular Point (Cont'd.) / 37 \\
                 7. Legendre Polynomials and Functions / 47 \\
                 8. Chebyshev, Hermite and Laguerre Polynomials / 57 \\
                 9. Bessel Functions / 64 \\
                 10. Hypergeometrie Functions / 75 \\
                 11. Piecewise Continuous and Periodic Functions / 83
                 \\
                 12. Orthogonal Functions and Polynomials / 90 \\
                 13. Orthogonal Functions and Polynomials (Cont'd.) / 95
                 \\
                 14. Boundary Value Problems / 104 \\
                 15. Boundary Value Problems (Cont'd.) / 109 \\
                 16. Green's Functions / 119 \\
                 17. Regular Perturbations / 129 \\
                 18. Singular Perturbations / 138 \\
                 19. Sturm--Liouville Problems / 145 \\
                 20. Eigenfunction Expansions / 157 \\
                 21. Eigenfunction Expansions (Cont'd.) / 163 \\
                 22. Convergence of the Fourier Series / 171 \\
                 23. Convergence of the Fourier Series (Cont'd.) / 176
                 \\
                 24. Fourier Series Solutions of Ordinary Differential
                 Equations / 187 \\
                 25. Partial Differential Equations / 194 \\
                 26. First-Order Partial Differential Equations / 202
                 \\
                 27. Solvable Partial Differential Equations / 210 \\
                 28. The Canonical Forms / 219 \\
                 29. The Method of Separation of Variables / 227 \\
                 30. The One-Dimensional Heat Equation / 234 \\
                 31. The One-Dimensional Heat Equation (Cont'd.) / 241
                 \\
                 32. The One-Dimensional Wave Equation / 249 \\
                 33. The One-Dimensional Wave Equation (Cont'd.) / 256
                 \\
                 34. Laplace Equation in Two Dimensions / 266 \\
                 35. Laplace Equation in Polar Coordinates / 275 \\
                 36. Two-Dimensional Heat Equation / 284 \\
                 37. Two-Dimensional Wave Equation / 292 \\
                 38. Laplace Equation in Three Dimensions / 300 \\
                 39. Laplace Equation in Three Dimensions (Cont'd.) /
                 306 \\
                 40. Nonhomogeneous Equations / 316 \\
                 41. Fourier Integral and Transforms / 323 \\
                 42. Fourier Integral and Transforms (Cont'd.) / 330 \\
                 43. Fourier Transform Method for Partial DEs / 338 \\
                 44. Fourier Transform Method for Partial DEs (Cont'd.)
                 / 344 \\
                 45. Laplace Transforms / 354 \\
                 46. Laplace Transforms (Cont'd.) / 361 \\
                 47. Laplace Transform Method for Ordinary DEs / 374 \\
                 48. Laplace Transform Method for Partial DEs / 384 \\
                 49. Well-Posed Problems / 394 \\
                 50. Verification of Solutions / 399 \\
                 References for Further Reading / 405 \\
                 Index / 407",
}


bibtype: article
filename: elefunt.bib
label: Alzer:2008:GFI
author: Horst Alzer
title: Gamma function inequalities
journal: j-NUMER-ALGORITHMS
volume: 49
number: 1--4
pages: 53--84
month: dec
monthnumber: 12
year: 2008
CODEN: NUALEG
ISSN: 1017-1398 (print), 1572-9265 (electronic) OR 10171398 (print), 15729265 (electronic)
ISSNL: 1017-1398 OR 10171398
bibdate: Mon May 17 14:08:26 MDT 2010
bibtimestamp: 2010.05.17 14:08:26 MDT
URL: http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=49&issue=1&spage=53
fjournal: Numerical Algorithms
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11075
rawauthor: Horst Alzer
rawtitle: Gamma function inequalities
bibsource:
http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1017-1398&volume=49&issue=1; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

entry:
@Article{Alzer:2008:GFI,
  author =       "Horst Alzer",
  title =        "Gamma function inequalities",
  journal =      j-NUMER-ALGORITHMS,
  volume =       "49",
  number =       "1--4",
  pages =        "53--84",
  month =        dec,
  year =         "2008",
  CODEN =        "NUALEG",
  ISSN =         "1017-1398 (print), 1572-9265 (electronic)",
  ISSN-L =       "1017-1398",
  bibdate =      "Mon May 17 14:08:26 MDT 2010",
  bibsource =    "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1017-1398&volume=49&issue=1;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=49&issue=1&spage=53",
  acknowledgement = ack-nhfb,
  fjournal =     "Numerical Algorithms",
  journal-URL =  "http://link.springer.com/journal/11075",
}


bibtype: article
filename: elefunt.bib
label: Ancarani:2008:DOC
author: L. U. Ancarani and G. Gasaneo
journal: j-J-MATH-PHYS
volume: 49
number: 6
pages: 063508
month: jun
monthnumber: 6
year: 2008
CODEN: JMAPAQ
DOI: https://doi.org/10.1063/1.2939395
ISSNL: 0022-2488 OR 00222488
bibdate: Wed Oct 26 09:06:03 MDT 2011
bibtimestamp: 2011.10.26 09:06:03 MDT
URL: http://jmp.aip.org/resource/1/jmapaq/v49/i6/p063508_s1
fjournal: Journal of Mathematical Physics
acknowledgement: ack-nhfb
journalURL: http://jmp.aip.org/
onlinedate: 20 June 2008
rawauthor: L. U. Ancarani and G. Gasaneo
title:
Derivatives of any order of the confluent hypergeometric function $_1 F_1 (a, b, z)$ with respect to the parameter $a$ or $b$

bibsource:
http://www.aip.org/ojs/jmp.html; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jmathphys2005.bib

rawtitle:
Derivatives of any order of the confluent hypergeometric function {$_1 F_1 (a, b, z)$} with respect to the parameter $a$ or $b$

entry:
@Article{Ancarani:2008:DOC,
  author =       "L. U. Ancarani and G. Gasaneo",
  title =        "Derivatives of any order of the confluent
                 hypergeometric function {$_1 F_1 (a, b, z)$} with
                 respect to the parameter $a$ or $b$",
  journal =      j-J-MATH-PHYS,
  volume =       "49",
  number =       "6",
  pages =        "063508",
  month =        jun,
  year =         "2008",
  CODEN =        "JMAPAQ",
  DOI =          "https://doi.org/10.1063/1.2939395",
  ISSN =         "0022-2488 (print), 1089-7658 (electronic), 1527-2427",
  ISSN-L =       "0022-2488",
  bibdate =      "Wed Oct 26 09:06:03 MDT 2011",
  bibsource =    "http://www.aip.org/ojs/jmp.html;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jmathphys2005.bib",
  URL =          "http://jmp.aip.org/resource/1/jmapaq/v49/i6/p063508_s1",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Mathematical Physics",
  journal-URL =  "http://jmp.aip.org/",
  onlinedate =   "20 June 2008",
  pagecount =    "16",
}


bibtype: article
filename: elefunt.bib
label: Borwein:2008:EBF
author: David Borwein and Jonathan M. Borwein and O-Yeat Chan
title: The evaluation of Bessel functions via exp--arc integrals
journal: j-J-MATH-ANAL-APPL
volume: 341
number: 1
pages: 478--500
month: may
monthnumber: 5
year: 2008
CODEN: JMANAK
DOI: https://doi.org/10.1016/j.jmaa.2007.10.003
ISSN: 0022-247X (print), 1096-0813 (electronic) OR 0022247X (print), 10960813 (electronic)
ISSNL: 0022-247X OR 0022247X
MRclass: 33C10 (33F05 65D20)
MRnumber: 2394100
MRreviewer: Richard B. Paris
bibdate: Thu Aug 11 10:27:38 2016
bibtimestamp: 2016.08.11 10:27:38 ???
fjournal: Journal of Mathematical Analysis and Applications
keywords: Bessel function; Exponential-hyperbolic expansions; Uniform series expansion
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/0022247X
rawauthor: David Borwein and Jonathan M. Borwein and O-Yeat Chan
rawtitle: The evaluation of {Bessel} functions via exp--arc integrals
ORCID: Borwein, Jonathan/0000-0002-1263-0646
bibsource:
https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

URL:
http://adsabs.harvard.edu/abs/2008JMAA..341..478B; http://docserver.carma.newcastle.edu.au/1231/; http://www.sciencedirect.com/science/article/pii/S0022247X07012346

abstract:
A standard method for computing values of Bessel functions has been to use the well-known ascending series for small argument, and to use an asymptotic series for large argument; with the choice of the series changing at some appropriate argument magnitude, depending on the number of digits required. In a recent paper, D. Borwein, J. Borwein, and R. Crandall [D. Borwein, J. M. Borwein, R. Crandall, Effective Laguerre asymptotics, preprint at http://locutus.cs.dal.ca:8088/archive/00000334/] derived a series for an ``exp-arc'' integral which gave rise to an absolutely convergent series for the J and I Bessel functions with integral order. Such series can be rapidly evaluated via recursion and elementary operations, and provide a viable alternative to the conventional ascending-asymptotic switching. In the present work, we extend the method to deal with Bessel functions of general (non-integral) order, as well as to deal with the Y and K Bessel functions.

rawabstract:
A standard method for computing values of Bessel functions has been to use the well-known ascending series for small argument, and to use an asymptotic series for large argument; with the choice of the series changing at some appropriate argument magnitude, depending on the number of digits required. In a recent paper, D. Borwein, J. Borwein, and R. Crandall [D. Borwein, J. M. Borwein, R. Crandall, Effective Laguerre asymptotics, preprint at http://locutus.cs.dal.ca:8088/archive/00000334/] derived a series for an ``exp-arc'' integral which gave rise to an absolutely convergent series for the J and I Bessel functions with integral order. Such series can be rapidly evaluated via recursion and elementary operations, and provide a viable alternative to the conventional ascending-asymptotic switching. In the present work, we extend the method to deal with Bessel functions of general (non-integral) order, as well as to deal with the Y and K Bessel functions.

entry:
@Article{Borwein:2008:EBF,
  author =       "David Borwein and Jonathan M. Borwein and O-Yeat
                 Chan",
  title =        "The evaluation of {Bessel} functions via exp--arc
                 integrals",
  journal =      j-J-MATH-ANAL-APPL,
  volume =       "341",
  number =       "1",
  pages =        "478--500",
  month =        may,
  year =         "2008",
  CODEN =        "JMANAK",
  DOI =          "https://doi.org/10.1016/j.jmaa.2007.10.003",
  ISSN =         "0022-247X (print), 1096-0813 (electronic)",
  ISSN-L =       "0022-247X",
  MRclass =      "33C10 (33F05 65D20)",
  MRnumber =     "2394100",
  MRreviewer =   "Richard B. Paris",
  bibdate =      "Thu Aug 11 10:27:38 2016",
  bibsource =    "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://adsabs.harvard.edu/abs/2008JMAA..341..478B;
                 http://docserver.carma.newcastle.edu.au/1231/;
                 http://www.sciencedirect.com/science/article/pii/S0022247X07012346",
  abstract =     "A standard method for computing values of Bessel
                 functions has been to use the well-known ascending
                 series for small argument, and to use an asymptotic
                 series for large argument; with the choice of the
                 series changing at some appropriate argument magnitude,
                 depending on the number of digits required. In a recent
                 paper, D. Borwein, J. Borwein, and R. Crandall [D.
                 Borwein, J. M. Borwein, R. Crandall, Effective Laguerre
                 asymptotics, preprint at
                 http://locutus.cs.dal.ca:8088/archive/00000334/]
                 derived a series for an ``exp-arc'' integral which gave
                 rise to an absolutely convergent series for the J and I
                 Bessel functions with integral order. Such series can
                 be rapidly evaluated via recursion and elementary
                 operations, and provide a viable alternative to the
                 conventional ascending-asymptotic switching. In the
                 present work, we extend the method to deal with Bessel
                 functions of general (non-integral) order, as well as
                 to deal with the Y and K Bessel functions.",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Mathematical Analysis and Applications",
  journal-URL =  "http://www.sciencedirect.com/science/journal/0022247X",
  keywords =     "Bessel function; Exponential-hyperbolic expansions;
                 Uniform series expansion",
  ORCID-numbers = "Borwein, Jonathan/0000-0002-1263-0646",
}


bibtype: article
filename: elefunt.bib
label: Borwein:2008:ELA
author: David Borwein and Jonathan M. Borwein and Richard E. Crandall
title: Effective Laguerre asymptotics
journal: j-SIAM-J-NUMER-ANAL
volume: 46
number: 6
pages: 3285--3312
year: 2008
CODEN: SJNAAM
DOI: https://doi.org/10.1137/07068031X
ISSN: 0036-1429 (print), 1095-7170 (electronic) OR 00361429 (print), 10957170 (electronic)
ISSNL: 0036-1429 OR 00361429
MRclass: 33C65 (30E20 34E05)
MRnumber: 2448665
MRreviewer: Yu-Qiu Zhao
bibdate: Wed Aug 10 11:09:47 2016
bibtimestamp: 2016.08.10 11:09:47 ???
URL: http://docserver.carma.newcastle.edu.au/334/
fjournal: SIAM Journal on Numerical Analysis
acknowledgement: ack-nhfb
journalURL: http://epubs.siam.org/sinum
rawauthor: David Borwein and Jonathan M. Borwein and Richard E. Crandall
rawtitle: Effective {Laguerre} asymptotics
ORCID: Borwein, Jonathan/0000-0002-1263-0646
bibsource:
https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

abstract:
It is known that the generalized Laguerre polynomials can enjoy subexponential growth for large primary index. In particular, for certain fixed parameter pairs (a, z) one has the large-n asymptotic behavior L-n((-a)) (-z) similar to C(a, z)(n)(-a)/2-1/ (4)e(2) root nz. We introduce a computationally motivated contour integral that allows efficient numerical Laguerre evaluations yet also leads to the complete asymptotic series over the full parameter domain of subexponential behavior. We present a fast algorithm for symbolic generation of the rather formidable expansion coefficients. Along the way we address the difficult problem of establishing effective (i. e., rigorous and explicit) error bounds on the general expansion. A primary tool for these developments is an ``exp-arc'' method giving a natural bridge between converging series and effective asymptotics.

rawabstract:
It is known that the generalized Laguerre polynomials can enjoy subexponential growth for large primary index. In particular, for certain fixed parameter pairs (a, z) one has the large-n asymptotic behavior L-n((-a)) (-z) similar to C(a, z)(n)(-a)/2-1/ (4)e(2) root nz. We introduce a computationally motivated contour integral that allows efficient numerical Laguerre evaluations yet also leads to the complete asymptotic series over the full parameter domain of subexponential behavior. We present a fast algorithm for symbolic generation of the rather formidable expansion coefficients. Along the way we address the difficult problem of establishing effective (i. e., rigorous and explicit) error bounds on the general expansion. A primary tool for these developments is an ``exp-arc'' method giving a natural bridge between converging series and effective asymptotics.

entry:
@Article{Borwein:2008:ELA,
  author =       "David Borwein and Jonathan M. Borwein and Richard E.
                 Crandall",
  title =        "Effective {Laguerre} asymptotics",
  journal =      j-SIAM-J-NUMER-ANAL,
  volume =       "46",
  number =       "6",
  pages =        "3285--3312",
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  bibdate =      "Wed Aug 10 11:09:47 2016",
  bibsource =    "https://www.math.utah.edu/pub/bibnet/authors/b/borwein-jonathan-m.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://docserver.carma.newcastle.edu.au/334/",
  abstract =     "It is known that the generalized Laguerre polynomials
                 can enjoy subexponential growth for large primary
                 index. In particular, for certain fixed parameter pairs
                 (a, z) one has the large-n asymptotic behavior
                 L-n((-a)) (-z) similar to C(a, z)(n)(-a)/2-1/ (4)e(2)
                 root nz. We introduce a computationally motivated
                 contour integral that allows efficient numerical
                 Laguerre evaluations yet also leads to the complete
                 asymptotic series over the full parameter domain of
                 subexponential behavior. We present a fast algorithm
                 for symbolic generation of the rather formidable
                 expansion coefficients. Along the way we address the
                 difficult problem of establishing effective (i. e.,
                 rigorous and explicit) error bounds on the general
                 expansion. A primary tool for these developments is an
                 ``exp-arc'' method giving a natural bridge between
                 converging series and effective asymptotics.",
  acknowledgement = ack-nhfb,
  fjournal =     "SIAM Journal on Numerical Analysis",
  journal-URL =  "http://epubs.siam.org/sinum",
  ORCID-numbers = "Borwein, Jonathan/0000-0002-1263-0646",
  researcherid-numbers = "Borwein, Jonathan/A-6082-2009",
  unique-id =    "Borwein:2008:ELA",
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bibtype: inproceedings
filename: elefunt.bib
label: Brisebarre:2008:EME
title: An Efficient Method for Evaluating Polynomial and Rational Function Approximations
crossref: IEEE:2008:ICA
pages: 233--238
year: 2008
DOI: https://doi.org/10.1109/ASAP.2008.4580185
bibdate: Mon Feb 10 07:28:25 2020
bibtimestamp: 2020.02.10 07:28:25 ???
acknowledgement: ack-nhfb
rawtitle: An Efficient Method for Evaluating Polynomial and Rational Function Approximations
author:
Nicolas Brisebarre and Sylvain Chevillard and Milos D. Ercegovac and Jean-Michel Muller and Serge Torres

rawauthor:
Nicolas Brisebarre and Sylvain Chevillard and Milo{\v{s}} D. Ercegovac and Jean-Michel Muller and Serge Torres

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@InProceedings{Brisebarre:2008:EME,
  author =       "Nicolas Brisebarre and Sylvain Chevillard and
                 Milo{\v{s}} D. Ercegovac and Jean-Michel Muller and
                 Serge Torres",
  title =        "An Efficient Method for Evaluating Polynomial and
                 Rational Function Approximations",
  crossref =     "IEEE:2008:ICA",
  pages =        "233--238",
  year =         "2008",
  DOI =          "https://doi.org/10.1109/ASAP.2008.4580185",
  bibdate =      "Mon Feb 10 07:28:25 2020",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: elefunt.bib
label: Chatterjee:2008:CNT
author: S. Chatterjee and D. Roy
title: A class of new transforms tailored for the hypergeometric series
journal: j-COMP-PHYS-COMM
volume: 179
number: 8
pages: 555--561
day: 15
month: oct
monthnumber: 10
year: 2008
CODEN: CPHCBZ
DOI: https://doi.org/10.1016/j.cpc.2008.05.001
ISSN: 0010-4655 (print), 1879-2944 (electronic) OR 00104655 (print), 18792944 (electronic)
ISSNL: 0010-4655 OR 00104655
bibdate: Thu Dec 01 09:09:57 2011
bibtimestamp: 2011.12.01 09:09:57 ???
fjournal: Computer Physics Communications
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00104655
rawauthor: S. Chatterjee and D. Roy
rawtitle: A class of new transforms tailored for the hypergeometric series
bibsource:
https://www.math.utah.edu/pub/tex/bib/compphyscomm2000.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

entry:
@Article{Chatterjee:2008:CNT,
  author =       "S. Chatterjee and D. Roy",
  title =        "A class of new transforms tailored for the
                 hypergeometric series",
  journal =      j-COMP-PHYS-COMM,
  volume =       "179",
  number =       "8",
  pages =        "555--561",
  day =          "15",
  month =        oct,
  year =         "2008",
  CODEN =        "CPHCBZ",
  DOI =          "https://doi.org/10.1016/j.cpc.2008.05.001",
  ISSN =         "0010-4655 (print), 1879-2944 (electronic)",
  ISSN-L =       "0010-4655",
  bibdate =      "Thu Dec 01 09:09:57 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/compphyscomm2000.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "Computer Physics Communications",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00104655",
  xxauthor =     "S. Charterjee and D. Roy",
}


bibtype: book
filename: elefunt.bib
label: Cuyt:2008:HCF
author: Annie Cuyt and Vigdis B. Petersen and Brigitte Verdonk and Haakon Waadeland and William B. Jones
title: Handbook of Continued Fractions for Special Functions
publisher: pub-SV
address: pub-SV:adr
pages: xx + 440
year: 2008
DOI: https://doi.org/10.1007/978-1-4020-6949-9
ISBN: 1-4020-6948-0 OR 1402069480
ISBN13: 978-1-4020-6948-2 OR 9781402069482
LCCN: QA295 .H275 2008
bibdate: Tue Jun 24 07:17:37 MDT 2008
bibtimestamp: 2008.06.24 07:17:37 MDT
acknowledgement: ack-nhfb
rawauthor: Annie Cuyt and Vigdis B. Petersen and Brigitte Verdonk and Haakon Waadeland and William B. Jones
rawtitle: Handbook of Continued Fractions for Special Functions
keywords:
applying the limit process; associated continued fraction; asymptotic series expansion; basic hypergeometric functions; canonical contraction; combination with property; complementary incomplete gamma function; complex error function; confluent hypergeometric series; continued fraction converges; continued fraction representations; fraction approximants; modified approximant; monic orthogonal polynomial sequence; normed field; nth approximant; nth denominator; nth numerator; nth tail; oval sequence theorem; parabola theorem; partial numerators; strong moment distribution function; successive approximants; truncation error bounds

TOC:
Preface / xi Notation / xiii 0 General considerations / 1 0.1 Part one / 1 0.2 Part two / 2 0.3 Part three / 2 Part I: Basic Theory 1 Basics / 9 1.1 Symbols and notation 1.2 Definitions / 10 1.3 Recurrence relations / 13 1.4 Equivalence transformations / 15 1.5 Contractions and extensions / 16 1.6 Continued fractions with prescribed approximants / 18 1.7 Connection between continued fractions and series / 19 1.8 Periodic and limit periodic continued fractions / 21 1.9 Tails of continued fractions / 23 1.10 Continued fractions over normed fields / 26 1.11 Generalisations of continued fractions / 28 2 Continued fraction representation of functions / 29 2.1 Symbols and notation / 29 2.2 Correspondence / 30 2.3 Families of continued fractions / 35 2.4 Correspondence of C-fractions / 39 2.5 Correspondence of P-fractions / 40 2.6 Correspondence of J-fractions and T-fractions / 41 2.7 Correspondence and three-term recurrences / 42 3 Convergence criteria / 45 3.1 Some classical theorems / 45 3.2 Convergence sets and value sets / 47 3.3 Parabola and oval theorems / 49 3.4 Correspondence and uniform convergence / 52 3.5 Periodic and limit periodic continued fractions / 53 3.6 Convergence and minimal solutions / 56 4 Pade approximants / 59 4.1 Definition and notation / 59 4.2 Fundamental properties / 60 4.3 Connection with regular C-fractions / 64 4.4 Connection with P-fractions / 65 4.5 Extension of the Pade table / 67 4.6 Connection with M-fractions and the M-table / 68 4.7 Convergence of Pade approximants / 70 4.8 Formal orthogonality property / 72 5 Moment theory and orthogonal functions / 77 5.1 Moment theory / 77 5.2 Stieltjes transforms / 85 5.3 Construction of solutions / 90 5.4 Orthogonal polynomials / 91 5.5 Monic orthogonal polynomials on $mathbb R$ and J-fractions / 92 5.6 Szego polynomials and PPC-fractions / 100 5.7 Orthogonal Laurent polynomials and APT-fractions / 102 Part II: Numerics 6 Continued fraction construction / 107 6.1 Regular C-fractions / 107 6.2 C-fractions / 113 6.3 S-fractions / 114 6.4 P-fractions / 114 6.5 J-fractions / 120 6.6 M-fractions / 122 6.7 Positive T-fractions / 124 6.8 Thiele fractions / 125 7 Truncation error bounds / 129 7.1 Parabola theorems / 129 7.2 The oval sequence theorem / 131 7.3 The interval sequence theorem / 136 7.4 Specific a priori bounds for S-fractions / 138 7.5 A posteriori truncation error bounds / 140 7.6 Tails and truncation error bounds / 143 7.7 Choice of modification / 143 8 Continued fraction evaluation / 149 8.1 The effect of finite precision arithmetic / 149 8.2 Evaluation of approximants / 152 8.3 The forward recurrence and minimal solutions / 154 8.4 Round-off error in the backward recurrence / 156 Part III: Special Functions 9 On tables and graphs / 163 9.1 Introduction / 163 9.2 Comparative tables / 163 9.3 Reliable graphs / 168 10 Mathematical constants / 175 10.1 Regular continued fractions / 175 10.2 Archimedes' constant, symbol $pi $ / 176 10.3 Euler's number, base of the natural logarithm / 178 10.4 Integer powers and roots of $pi $ and $e$ / 180 10.5 The natural logarithm, $ln (2)$ / 181 10.6 Pythagoras' constant, the square root of two / 183 10.7 The cube root of two / 183 10.8 Euler's constant, symbol $gamma $ / 185 10.9 Golden ratio, symbol $phi $ / 185 10.10 The rabbit constant, symbol $rho $ / 186 10.11 Apery's constant, $zeta (3)$ / 188 10.12 Catalan's constant, symbol $C$ / 189 10.13 Gompertz' constant, symbol $G$ / 190 10.14 Khinchin's constant, symbol $K$ / 190 11 Elementary functions / 193 11.1 The exponential function / 193 11.2 The natural logarithm / 196 11.3 Trigonometric functions / 200 11.4 Inverse trigonometric functions / 204 11.5 Hyperbolic functions / 210 11.6 Inverse hyperbolic functions / 213 11.7 The power function / 217 12 Gamma function and related functions / 221 12.1 Gamma function / 221 12.2 Binet function / 224 12.3 Polygamma functions / 229 12.4 Trigamma function / 232 12.5 Tetragamma function / 235 12.6 Incomplete gamma functions / 238 13 Error function and related integrals / 253 13.1 Error function and Dawson's integral / 253 13.2 Complementary and complex error function / 261 13.3 Repeated integrals / 268 13.4 Fresnel integrals / 269 14 Exponential integrals and related functions / 275 14.7 Exponential integrals / 275 14.2 Related functions / 285 15 Hypergeometric functions / 291 15.1 Definition and basic properties / 291 15.2 Stieltjes transform / 295 15.3 Continued fraction representations / 295 15.4 Pade approximants / 309 15.5 Monotonicity properties / 313 15.6 Hypergeometric series $_pF_q$ / 315 16 Confluent hypergeometric functions / 319 16.1 Kummer functions / 319 16.2 Confluent hypergeometric series $_2F_0$ / 330 16.3 Confluent hypergeometric limit function / 333 16.4 Whittaker functions / 334 16.5 Parabolic cylinder functions / 337 17 Bessel functions / 334 17.7 Bessel functions / 334 17.2 Modified Bessel functions / 356 18 Probability functions / 371 18.1 Definitions and elementary properties / 371 18.2 Normal and log-normal distributions / 373 18.3 Repeated integrals / 377 18.4 Gamma and chi-square distribution / 378 18.5 Beta, $F$- and Student's $t$-distributions / 382 19 Basic hypergeometric functions / 391 19.1 Definition and basic properties / 391 19.2 Continued fraction representations / 395 19.3 Higher order basic hypergeometric functions / 399 Bibliography / 401 Index / 421

shortTOC:
General considerations Part 1, Basic Theory 1. Basics 2. Continued fraction representation of functions 3. Convergence criteria 4. Pade approximants 5. Moment theory and orthogonal functions Part 2, Numerics 6. Continued fraction construction 7. Truncation error bounds 8. Continued fraction evaluation Part 3, Special Functions 9. On tables and graphs 10. Mathematical constants 11. Elementary functions 12. Gamma function and related functions 13. Error function and related integrals 14. Exponential integrals and related functions 15. Hypergeometric functions 16. Confluent hypergeometric functions, 17. Bessel functions 18. Probability functions 19. Basic hypergeometric functions

rawshortTOC:
General considerations \\ Part 1, Basic Theory \\ 1. Basics \\ 2. Continued fraction representation of functions \\ 3. Convergence criteria \\ 4. Pade approximants \\ 5. Moment theory and orthogonal functions \\ Part 2, Numerics \\ 6. Continued fraction construction \\ 7. Truncation error bounds \\ 8. Continued fraction evaluation \\ Part 3, Special Functions \\ 9. On tables and graphs \\ 10. Mathematical constants \\ 11. Elementary functions \\ 12. Gamma function and related functions \\ 13. Error function and related integrals \\ 14. Exponential integrals and related functions \\ 15. Hypergeometric functions \\ 16. Confluent hypergeometric functions, \\ 17. Bessel functions \\ 18. Probability functions \\ 19. Basic hypergeometric functions

rawTOC:
Preface / xi \\ Notation / xiii \\ 0 General considerations \\ / 1 \\ 0.1 Part one / 1 \\ 0.2 Part two / 2 \\ 0.3 Part three / 2 \\ \\ Part I: Basic Theory \\ \\ 1 Basics / 9 \\ 1.1 Symbols and notation \\ 1.2 Definitions / 10 \\ 1.3 Recurrence relations / 13 \\ 1.4 Equivalence transformations / 15 \\ 1.5 Contractions and extensions / 16 \\ 1.6 Continued fractions with prescribed approximants / 18 \\ 1.7 Connection between continued fractions and series / 19 \\ 1.8 Periodic and limit periodic continued fractions / 21 \\ 1.9 Tails of continued fractions / 23 \\ 1.10 Continued fractions over normed fields / 26 \\ 1.11 Generalisations of continued fractions / 28 \\ \\ 2 Continued fraction representation of functions / 29 \\ 2.1 Symbols and notation / 29 \\ 2.2 Correspondence / 30 \\ 2.3 Families of continued fractions / 35 \\ 2.4 Correspondence of C-fractions / 39 \\ 2.5 Correspondence of P-fractions / 40 \\ 2.6 Correspondence of J-fractions and T-fractions / 41 \\ 2.7 Correspondence and three-term recurrences / 42 \\ \\ 3 Convergence criteria / 45 \\ 3.1 Some classical theorems / 45 \\ 3.2 Convergence sets and value sets / 47 \\ 3.3 Parabola and oval theorems / 49 \\ 3.4 Correspondence and uniform convergence / 52 \\ 3.5 Periodic and limit periodic continued fractions / 53 \\ 3.6 Convergence and minimal solutions / 56 \\ \\ 4 Pad{\'e} approximants / 59 \\ 4.1 Definition and notation / 59 \\ 4.2 Fundamental properties / 60 \\ 4.3 Connection with regular C-fractions / 64 \\ 4.4 Connection with P-fractions / 65 \\ 4.5 Extension of the Pad{\'e} table / 67 \\ 4.6 Connection with M-fractions and the M-table / 68 \\ 4.7 Convergence of Pad{\'e} approximants / 70 \\ 4.8 Formal orthogonality property / 72 \\ \\ 5 Moment theory and orthogonal functions / 77 \\ 5.1 Moment theory / 77 \\ 5.2 Stieltjes transforms / 85 \\ 5.3 Construction of solutions / 90 \\ 5.4 Orthogonal polynomials / 91 \\ 5.5 Monic orthogonal polynomials on $\mathbb{R}$ and J-fractions / 92 \\ 5.6 Szeg{\H{o}} polynomials and PPC-fractions / 100 \\ 5.7 Orthogonal Laurent polynomials and APT-fractions / 102 \\ \\ Part II: Numerics \\ \\ 6 Continued fraction construction / 107 \\ 6.1 Regular C-fractions / 107 \\ 6.2 C-fractions / 113 \\ 6.3 S-fractions / 114 \\ 6.4 P-fractions / 114 \\ 6.5 J-fractions / 120 \\ 6.6 M-fractions / 122 \\ 6.7 Positive T-fractions / 124 \\ 6.8 Thiele fractions / 125 \\ \\ 7 Truncation error bounds / 129 \\ 7.1 Parabola theorems / 129 \\ 7.2 The oval sequence theorem / 131 \\ 7.3 The interval sequence theorem / 136 \\ 7.4 Specific a priori bounds for S-fractions / 138 \\ 7.5 A posteriori truncation error bounds / 140 \\ 7.6 Tails and truncation error bounds / 143 \\ 7.7 Choice of modification / 143 \\ \\ 8 Continued fraction evaluation / 149 \\ 8.1 The effect of finite precision arithmetic / 149 \\ 8.2 Evaluation of approximants / 152 \\ 8.3 The forward recurrence and minimal solutions / 154 \\ 8.4 Round-off error in the backward recurrence / 156 \\ \\ Part III: Special Functions \\ \\ 9 On tables and graphs / 163 \\ 9.1 Introduction / 163 \\ 9.2 Comparative tables / 163 \\ 9.3 Reliable graphs / 168 \\ \\ 10 Mathematical constants / 175 \\ 10.1 Regular continued fractions / 175 \\ 10.2 Archimedes' constant, symbol $\pi$ / 176 \\ 10.3 Euler's number, base of the natural logarithm / 178 \\ 10.4 Integer powers and roots of $\pi$ and $e$ / 180 \\ 10.5 The natural logarithm, $\ln(2)$ / 181 \\ 10.6 Pythagoras' constant, the square root of two / 183 \\ 10.7 The cube root of two / 183 \\ 10.8 Euler's constant, symbol $\gamma$ / 185 \\ 10.9 Golden ratio, symbol $\phi$ / 185 \\ 10.10 The rabbit constant, symbol $\rho$ / 186 \\ 10.11 Ap{\'e}ry's constant, $\zeta(3)$ / 188 \\ 10.12 Catalan's constant, symbol $C$ / 189 \\ 10.13 Gompertz' constant, symbol $G$ / 190 \\ 10.14 Khinchin's constant, symbol $K$ / 190 \\ \\ 11 Elementary functions / 193 \\ 11.1 The exponential function / 193 \\ 11.2 The natural logarithm / 196 \\ 11.3 Trigonometric functions / 200 \\ 11.4 Inverse trigonometric functions / 204 \\ 11.5 Hyperbolic functions / 210 \\ 11.6 Inverse hyperbolic functions / 213 \\ 11.7 The power function / 217 \\ \\ 12 Gamma function and related functions / 221 \\ 12.1 Gamma function / 221 \\ 12.2 Binet function / 224 \\ 12.3 Polygamma functions / 229 \\ 12.4 Trigamma function / 232 \\ 12.5 Tetragamma function / 235 \\ 12.6 Incomplete gamma functions / 238 \\ \\ 13 Error function and related integrals / 253 \\ 13.1 Error function and Dawson's integral / 253 \\ 13.2 Complementary and complex error function / 261 \\ 13.3 Repeated integrals / 268 \\ 13.4 Fresnel integrals / 269 \\ \\ 14 Exponential integrals and related functions / 275 \\ 14.7 Exponential integrals / 275 \\ 14.2 Related functions / 285 \\ \\ 15 Hypergeometric functions / 291 \\ 15.1 Definition and basic properties / 291 \\ 15.2 Stieltjes transform / 295 \\ 15.3 Continued fraction representations / 295 \\ 15.4 Pad{\'e} approximants / 309 \\ 15.5 Monotonicity properties / 313 \\ 15.6 Hypergeometric series $_pF_q$ / 315 \\ \\ 16 Confluent hypergeometric functions / 319 \\ 16.1 Kummer functions / 319 \\ 16.2 Confluent hypergeometric series $_2F_0$ / 330 \\ 16.3 Confluent hypergeometric limit function / 333 \\ 16.4 Whittaker functions / 334 \\ 16.5 Parabolic cylinder functions / 337 \\ \\ 17 Bessel functions / 334 \\ 17.7 Bessel functions / 334 \\ 17.2 Modified Bessel functions / 356 \\ \\ 18 Probability functions / 371 \\ 18.1 Definitions and elementary properties / 371 \\ 18.2 Normal and log-normal distributions / 373 \\ 18.3 Repeated integrals / 377 \\ 18.4 Gamma and chi-square distribution / 378 \\ 18.5 Beta, $F$- and Student's $t$-distributions / 382 \\ \\ 19 Basic hypergeometric functions / 391 \\ 19.1 Definition and basic properties / 391 \\ 19.2 Continued fraction representations / 395 \\ 19.3 Higher order basic hypergeometric functions / 399 \\ \\ Bibliography / 401 \\ \\ Index / 421

entry:
@Book{Cuyt:2008:HCF,
  author =       "Annie Cuyt and Vigdis B. Petersen and Brigitte Verdonk
                 and Haakon Waadeland and William B. Jones",
  title =        "Handbook of Continued Fractions for Special
                 Functions",
  publisher =    pub-SV,
  address =      pub-SV:adr,
  pages =        "xx + 440",
  year =         "2008",
  DOI =          "https://doi.org/10.1007/978-1-4020-6949-9",
  ISBN =         "1-4020-6948-0",
  ISBN-13 =      "978-1-4020-6948-2",
  LCCN =         "QA295 .H275 2008",
  bibdate =      "Tue Jun 24 07:17:37 MDT 2008",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/master.bib;
                 z3950.loc.gov:7090/Voyager",
  acknowledgement = ack-nhfb,
  keywords =     "applying the limit process; associated continued
                 fraction; asymptotic series expansion; basic
                 hypergeometric functions; canonical contraction;
                 combination with property; complementary incomplete
                 gamma function; complex error function; confluent
                 hypergeometric series; continued fraction converges;
                 continued fraction representations; fraction
                 approximants; modified approximant; monic orthogonal
                 polynomial sequence; normed field; nth approximant; nth
                 denominator; nth numerator; nth tail; oval sequence
                 theorem; parabola theorem; partial numerators; strong
                 moment distribution function; successive approximants;
                 truncation error bounds",
  shorttableofcontents = "General considerations \\
                 Part 1, Basic Theory \\
                 1. Basics \\
                 2. Continued fraction representation of functions \\
                 3. Convergence criteria \\
                 4. Pade approximants \\
                 5. Moment theory and orthogonal functions \\
                 Part 2, Numerics \\
                 6. Continued fraction construction \\
                 7. Truncation error bounds \\
                 8. Continued fraction evaluation \\
                 Part 3, Special Functions \\
                 9. On tables and graphs \\
                 10. Mathematical constants \\
                 11. Elementary functions \\
                 12. Gamma function and related functions \\
                 13. Error function and related integrals \\
                 14. Exponential integrals and related functions \\
                 15. Hypergeometric functions \\
                 16. Confluent hypergeometric functions, \\
                 17. Bessel functions \\
                 18. Probability functions \\
                 19. Basic hypergeometric functions",
  tableofcontents = "Preface / xi \\
                 Notation / xiii \\
                 0 General considerations \\
                 / 1 \\
                 0.1 Part one / 1 \\
                 0.2 Part two / 2 \\
                 0.3 Part three / 2 \\
                 \\
                 Part I: Basic Theory \\
                 \\
                 1 Basics / 9 \\
                 1.1 Symbols and notation \\
                 1.2 Definitions / 10 \\
                 1.3 Recurrence relations / 13 \\
                 1.4 Equivalence transformations / 15 \\
                 1.5 Contractions and extensions / 16 \\
                 1.6 Continued fractions with prescribed approximants /
                 18 \\
                 1.7 Connection between continued fractions and series /
                 19 \\
                 1.8 Periodic and limit periodic continued fractions /
                 21 \\
                 1.9 Tails of continued fractions / 23 \\
                 1.10 Continued fractions over normed fields / 26 \\
                 1.11 Generalisations of continued fractions / 28 \\
                 \\
                 2 Continued fraction representation of functions / 29
                 \\
                 2.1 Symbols and notation / 29 \\
                 2.2 Correspondence / 30 \\
                 2.3 Families of continued fractions / 35 \\
                 2.4 Correspondence of C-fractions / 39 \\
                 2.5 Correspondence of P-fractions / 40 \\
                 2.6 Correspondence of J-fractions and T-fractions / 41
                 \\
                 2.7 Correspondence and three-term recurrences / 42 \\
                 \\
                 3 Convergence criteria / 45 \\
                 3.1 Some classical theorems / 45 \\
                 3.2 Convergence sets and value sets / 47 \\
                 3.3 Parabola and oval theorems / 49 \\
                 3.4 Correspondence and uniform convergence / 52 \\
                 3.5 Periodic and limit periodic continued fractions /
                 53 \\
                 3.6 Convergence and minimal solutions / 56 \\
                 \\
                 4 Pad{\'e} approximants / 59 \\
                 4.1 Definition and notation / 59 \\
                 4.2 Fundamental properties / 60 \\
                 4.3 Connection with regular C-fractions / 64 \\
                 4.4 Connection with P-fractions / 65 \\
                 4.5 Extension of the Pad{\'e} table / 67 \\
                 4.6 Connection with M-fractions and the M-table / 68
                 \\
                 4.7 Convergence of Pad{\'e} approximants / 70 \\
                 4.8 Formal orthogonality property / 72 \\
                 \\
                 5 Moment theory and orthogonal functions / 77 \\
                 5.1 Moment theory / 77 \\
                 5.2 Stieltjes transforms / 85 \\
                 5.3 Construction of solutions / 90 \\
                 5.4 Orthogonal polynomials / 91 \\
                 5.5 Monic orthogonal polynomials on $\mathbb{R}$ and
                 J-fractions / 92 \\
                 5.6 Szeg{\H{o}} polynomials and PPC-fractions / 100 \\
                 5.7 Orthogonal Laurent polynomials and APT-fractions /
                 102 \\
                 \\
                 Part II: Numerics \\
                 \\
                 6 Continued fraction construction / 107 \\
                 6.1 Regular C-fractions / 107 \\
                 6.2 C-fractions / 113 \\
                 6.3 S-fractions / 114 \\
                 6.4 P-fractions / 114 \\
                 6.5 J-fractions / 120 \\
                 6.6 M-fractions / 122 \\
                 6.7 Positive T-fractions / 124 \\
                 6.8 Thiele fractions / 125 \\
                 \\
                 7 Truncation error bounds / 129 \\
                 7.1 Parabola theorems / 129 \\
                 7.2 The oval sequence theorem / 131 \\
                 7.3 The interval sequence theorem / 136 \\
                 7.4 Specific a priori bounds for S-fractions / 138 \\
                 7.5 A posteriori truncation error bounds / 140 \\
                 7.6 Tails and truncation error bounds / 143 \\
                 7.7 Choice of modification / 143 \\
                 \\
                 8 Continued fraction evaluation / 149 \\
                 8.1 The effect of finite precision arithmetic / 149 \\
                 8.2 Evaluation of approximants / 152 \\
                 8.3 The forward recurrence and minimal solutions / 154
                 \\
                 8.4 Round-off error in the backward recurrence / 156
                 \\
                 \\
                 Part III: Special Functions \\
                 \\
                 9 On tables and graphs / 163 \\
                 9.1 Introduction / 163 \\
                 9.2 Comparative tables / 163 \\
                 9.3 Reliable graphs / 168 \\
                 \\
                 10 Mathematical constants / 175 \\
                 10.1 Regular continued fractions / 175 \\
                 10.2 Archimedes' constant, symbol $\pi$ / 176 \\
                 10.3 Euler's number, base of the natural logarithm /
                 178 \\
                 10.4 Integer powers and roots of $\pi$ and $e$ / 180
                 \\
                 10.5 The natural logarithm, $\ln(2)$ / 181 \\
                 10.6 Pythagoras' constant, the square root of two / 183
                 \\
                 10.7 The cube root of two / 183 \\
                 10.8 Euler's constant, symbol $\gamma$ / 185 \\
                 10.9 Golden ratio, symbol $\phi$ / 185 \\
                 10.10 The rabbit constant, symbol $\rho$ / 186 \\
                 10.11 Ap{\'e}ry's constant, $\zeta(3)$ / 188 \\
                 10.12 Catalan's constant, symbol $C$ / 189 \\
                 10.13 Gompertz' constant, symbol $G$ / 190 \\
                 10.14 Khinchin's constant, symbol $K$ / 190 \\
                 \\
                 11 Elementary functions / 193 \\
                 11.1 The exponential function / 193 \\
                 11.2 The natural logarithm / 196 \\
                 11.3 Trigonometric functions / 200 \\
                 11.4 Inverse trigonometric functions / 204 \\
                 11.5 Hyperbolic functions / 210 \\
                 11.6 Inverse hyperbolic functions / 213 \\
                 11.7 The power function / 217 \\
                 \\
                 12 Gamma function and related functions / 221 \\
                 12.1 Gamma function / 221 \\
                 12.2 Binet function / 224 \\
                 12.3 Polygamma functions / 229 \\
                 12.4 Trigamma function / 232 \\
                 12.5 Tetragamma function / 235 \\
                 12.6 Incomplete gamma functions / 238 \\
                 \\
                 13 Error function and related integrals / 253 \\
                 13.1 Error function and Dawson's integral / 253 \\
                 13.2 Complementary and complex error function / 261 \\
                 13.3 Repeated integrals / 268 \\
                 13.4 Fresnel integrals / 269 \\
                 \\
                 14 Exponential integrals and related functions / 275
                 \\
                 14.7 Exponential integrals / 275 \\
                 14.2 Related functions / 285 \\
                 \\
                 15 Hypergeometric functions / 291 \\
                 15.1 Definition and basic properties / 291 \\
                 15.2 Stieltjes transform / 295 \\
                 15.3 Continued fraction representations / 295 \\
                 15.4 Pad{\'e} approximants / 309 \\
                 15.5 Monotonicity properties / 313 \\
                 15.6 Hypergeometric series $_pF_q$ / 315 \\
                 \\
                 16 Confluent hypergeometric functions / 319 \\
                 16.1 Kummer functions / 319 \\
                 16.2 Confluent hypergeometric series $_2F_0$ / 330 \\
                 16.3 Confluent hypergeometric limit function / 333 \\
                 16.4 Whittaker functions / 334 \\
                 16.5 Parabolic cylinder functions / 337 \\
                 \\
                 17 Bessel functions / 334 \\
                 17.7 Bessel functions / 334 \\
                 17.2 Modified Bessel functions / 356 \\
                 \\
                 18 Probability functions / 371 \\
                 18.1 Definitions and elementary properties / 371 \\
                 18.2 Normal and log-normal distributions / 373 \\
                 18.3 Repeated integrals / 377 \\
                 18.4 Gamma and chi-square distribution / 378 \\
                 18.5 Beta, $F$- and Student's $t$-distributions / 382
                 \\
                 \\
                 19 Basic hypergeometric functions / 391 \\
                 19.1 Definition and basic properties / 391 \\
                 19.2 Continued fraction representations / 395 \\
                 19.3 Higher order basic hypergeometric functions / 399
                 \\
                 \\
                 Bibliography / 401 \\
                 \\
                 Index / 421",
}


bibtype: article
filename: elefunt.bib
label: Dyer:2008:CCI
author: J. S. Dyer and S. A. Dyer
title: Corrections to, and comments on, ``An improved approximation for the Gaussian $Q$-Function''
journal: j-IEEE-COMMUN-LET
volume: 12
number: 4
pages: 231--231
month: apr
monthnumber: 4
year: 2008
CODEN: ICLEF6
DOI: https://doi.org/10.1109/lcomm.2008.080009
ISSN: 1089-7798 (print), 1558-2558 (electronic) OR 10897798 (print), 15582558 (electronic)
ISSNL: 1089-7798 OR 10897798
bibdate: Sat Dec 16 18:08:34 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2017.12.16 18:08:34 ???
note: See cite Karagiannidis:2007:IAG.
URL: http://ieeexplore.ieee.org/document/4489650/
fjournal: IEEE Communications Letters
acknowledgement: ack-nhfb
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=4234
rawauthor: J. S. Dyer and S. A. Dyer
rawnote: See \cite{Karagiannidis:2007:IAG}.
rawtitle: Corrections to, and comments on, {``An improved approximation for the Gaussian $Q$-Function''}
entry:
@Article{Dyer:2008:CCI,
  author =       "J. S. Dyer and S. A. Dyer",
  title =        "Corrections to, and comments on, {``An improved
                 approximation for the Gaussian $Q$-Function''}",
  journal =      j-IEEE-COMMUN-LET,
  volume =       "12",
  number =       "4",
  pages =        "231--231",
  month =        apr,
  year =         "2008",
  CODEN =        "ICLEF6",
  DOI =          "https://doi.org/10.1109/lcomm.2008.080009",
  ISSN =         "1089-7798 (print), 1558-2558 (electronic)",
  ISSN-L =       "1089-7798",
  bibdate =      "Sat Dec 16 18:08:34 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  note =         "See \cite{Karagiannidis:2007:IAG}.",
  URL =          "http://ieeexplore.ieee.org/document/4489650/",
  acknowledgement = ack-nhfb,
  fjournal =     "IEEE Communications Letters",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=4234",
}


bibtype: article
filename: elefunt.bib
label: Elbert:2008:ZCE
author: Arpad Elbert and Andrea Laforgia
title: The zeros of the complementary error function
journal: j-NUMER-ALGORITHMS
volume: 49
number: 1--4
pages: 153--157
month: dec
monthnumber: 12
year: 2008
CODEN: NUALEG
ISSN: 1017-1398 (print), 1572-9265 (electronic) OR 10171398 (print), 15729265 (electronic)
ISSNL: 1017-1398 OR 10171398
bibdate: Mon May 17 14:08:26 MDT 2010
bibtimestamp: 2010.05.17 14:08:26 MDT
URL: http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=49&issue=1&spage=153
fjournal: Numerical Algorithms
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11075
rawauthor: {\'A}rp{\'a}d Elbert and Andrea Laforgia
rawtitle: The zeros of the complementary error function
bibsource:
http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1017-1398&volume=49&issue=1; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

entry:
@Article{Elbert:2008:ZCE,
  author =       "{\'A}rp{\'a}d Elbert and Andrea Laforgia",
  title =        "The zeros of the complementary error function",
  journal =      j-NUMER-ALGORITHMS,
  volume =       "49",
  number =       "1--4",
  pages =        "153--157",
  month =        dec,
  year =         "2008",
  CODEN =        "NUALEG",
  ISSN =         "1017-1398 (print), 1572-9265 (electronic)",
  ISSN-L =       "1017-1398",
  bibdate =      "Mon May 17 14:08:26 MDT 2010",
  bibsource =    "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1017-1398&volume=49&issue=1;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=49&issue=1&spage=153",
  acknowledgement = ack-nhfb,
  fjournal =     "Numerical Algorithms",
  journal-URL =  "http://link.springer.com/journal/11075",
}


bibtype: article
filename: elefunt.bib
label: Gabutti:2008:EQG
author: Bruno Gabutti and Giampietro Allasia
title: Evaluation of $q$-gamma function and $q$-analogues by iterative algorithms
journal: j-NUMER-ALGORITHMS
volume: 49
number: 1--4
pages: 159--168
month: dec
monthnumber: 12
year: 2008
CODEN: NUALEG
ISSN: 1017-1398 (print), 1572-9265 (electronic) OR 10171398 (print), 15729265 (electronic)
ISSNL: 1017-1398 OR 10171398
bibdate: Mon May 17 15:07:19 MDT 2010
bibtimestamp: 2010.05.17 15:07:19 MDT
URL: http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=49&issue=1&spage=159
fjournal: Numerical Algorithms
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11075
rawauthor: Bruno Gabutti and Giampietro Allasia
rawtitle: Evaluation of $q$-gamma function and $q$-analogues by iterative algorithms
bibsource:
http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1017-1398&volume=49&issue=1; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

entry:
@Article{Gabutti:2008:EQG,
  author =       "Bruno Gabutti and Giampietro Allasia",
  title =        "Evaluation of $q$-gamma function and $q$-analogues by
                 iterative algorithms",
  journal =      j-NUMER-ALGORITHMS,
  volume =       "49",
  number =       "1--4",
  pages =        "159--168",
  month =        dec,
  year =         "2008",
  CODEN =        "NUALEG",
  ISSN =         "1017-1398 (print), 1572-9265 (electronic)",
  ISSN-L =       "1017-1398",
  bibdate =      "Mon May 17 15:07:19 MDT 2010",
  bibsource =    "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1017-1398&volume=49&issue=1;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=49&issue=1&spage=159",
  acknowledgement = ack-nhfb,
  fjournal =     "Numerical Algorithms",
  journal-URL =  "http://link.springer.com/journal/11075",
}


bibtype: article
filename: elefunt.bib
label: Gautschi:2008:LGW
author: Walter Gautschi and Carla Giordano
title: Luigi Gatteschi's work on asymptotics of special functions and their zeros
journal: j-NUMER-ALGORITHMS
volume: 49
number: 1--4
pages: 11--31
month: dec
monthnumber: 12
year: 2008
CODEN: NUALEG
ISSN: 1017-1398 (print), 1572-9265 (electronic) OR 10171398 (print), 15729265 (electronic)
ISSNL: 1017-1398 OR 10171398
bibdate: Mon May 17 14:08:26 MDT 2010
bibtimestamp: 2010.05.17 14:08:26 MDT
URL: http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=49&issue=1&spage=11
fjournal: Numerical Algorithms
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11075
rawauthor: Walter Gautschi and Carla Giordano
rawtitle: {Luigi Gatteschi}'s work on asymptotics of special functions and their zeros
entry:
@Article{Gautschi:2008:LGW,
  author =       "Walter Gautschi and Carla Giordano",
  title =        "{Luigi Gatteschi}'s work on asymptotics of special
                 functions and their zeros",
  journal =      j-NUMER-ALGORITHMS,
  volume =       "49",
  number =       "1--4",
  pages =        "11--31",
  month =        dec,
  year =         "2008",
  CODEN =        "NUALEG",
  ISSN =         "1017-1398 (print), 1572-9265 (electronic)",
  ISSN-L =       "1017-1398",
  bibdate =      "Mon May 17 14:08:26 MDT 2010",
  bibsource =    "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1017-1398&volume=49&issue=1;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=49&issue=1&spage=11",
  acknowledgement = ack-nhfb,
  fjournal =     "Numerical Algorithms",
  journal-URL =  "http://link.springer.com/journal/11075",
}


bibtype: article
filename: elefunt.bib
label: Harris:2008:IBG
author: Frank E. Harris
title: Incomplete Bessel, generalized incomplete gamma, or leaky aquifer functions
journal: j-J-COMPUT-APPL-MATH
volume: 215
number: 1
pages: 260--269
year: 2008
CODEN: JCAMDI
DOI: https://doi.org/10.1016/j.cam.2007.04.008
ISSN: 0377-0427 (print), 1879-1778 (electronic) OR 03770427 (print), 18791778 (electronic)
ISSNL: 0377-0427 OR 03770427
MRclass: 33B10 (33C10 41A58); 33B20
MRnumber: 2400632 (2009d:33003)
MRreviewer: Necdet Batir
bibdate: Wed Dec 4 07:03:09 2013
bibtimestamp: 2013.12.04 07:03:09 ???
URL: http://www.sciencedirect.com/science/article/pii/S0377042707002014
fjournal: Journal of Computational and Applied Mathematics
ZMnumber: Zbl 1135.33002
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03770427
rawauthor: Frank E. Harris
rawtitle: Incomplete {Bessel}, generalized incomplete gamma, or leaky aquifer functions
bibsource:
https://www.math.utah.edu/pub/bibnet/authors/h/harris-frank-e.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

entry:
@Article{Harris:2008:IBG,
  author =       "Frank E. Harris",
  title =        "Incomplete {Bessel}, generalized incomplete gamma, or
                 leaky aquifer functions",
  journal =      j-J-COMPUT-APPL-MATH,
  volume =       "215",
  number =       "1",
  pages =        "260--269",
  year =         "2008",
  CODEN =        "JCAMDI",
  DOI =          "https://doi.org/10.1016/j.cam.2007.04.008",
  ISSN =         "0377-0427 (print), 1879-1778 (electronic)",
  ISSN-L =       "0377-0427",
  MRclass =      "33B10 (33C10 41A58); 33B20",
  MRnumber =     "2400632 (2009d:33003)",
  MRreviewer =   "Necdet Batir",
  bibdate =      "Wed Dec 4 07:03:09 2013",
  bibsource =    "https://www.math.utah.edu/pub/bibnet/authors/h/harris-frank-e.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0377042707002014",
  ZMnumber =     "Zbl 1135.33002",
  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: Isukapalli:2008:ATA
author: Yogananda Isukapalli and Bhaskar D. Rao
title: An Analytically Tractable Approximation for the Gaussian $Q$-Function
journal: j-IEEE-COMMUN-LET
volume: 12
number: 9
pages: 669--671
month: sep
monthnumber: 9
year: 2008
CODEN: ICLEF6
DOI: https://doi.org/10.1109/lcomm.2008.080815
ISSN: 1089-7798 (print), 1558-2558 (electronic) OR 10897798 (print), 15582558 (electronic)
ISSNL: 1089-7798 OR 10897798
bibdate: Sat Dec 16 16:44:42 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2017.12.16 16:44:42 ???
fjournal: IEEE Communications Letters
acknowledgement: ack-nhfb
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=4234
rawauthor: Yogananda Isukapalli and Bhaskar D. Rao
rawtitle: An Analytically Tractable Approximation for the {Gaussian} {$Q$}-Function
entry:
@Article{Isukapalli:2008:ATA,
  author =       "Yogananda Isukapalli and Bhaskar D. Rao",
  title =        "An Analytically Tractable Approximation for the
                 {Gaussian} {$Q$}-Function",
  journal =      j-IEEE-COMMUN-LET,
  volume =       "12",
  number =       "9",
  pages =        "669--671",
  month =        sep,
  year =         "2008",
  CODEN =        "ICLEF6",
  DOI =          "https://doi.org/10.1109/lcomm.2008.080815",
  ISSN =         "1089-7798 (print), 1558-2558 (electronic)",
  ISSN-L =       "1089-7798",
  bibdate =      "Sat Dec 16 16:44:42 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "IEEE Communications Letters",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=4234",
}


bibtype: book
filename: elefunt.bib
label: Jeffrey:2008:HMF
author: Alan Jeffrey and Hui-Hui Dai
title: Handbook of Mathematical Formulas and Integrals
publisher: pub-ELSEVIER-ACADEMIC
address: pub-ELSEVIER-ACADEMIC:adr
edition: Fourth
pages: xlv + 541
year: 2008
ISBN: 0-12-374288-9 (paperback), 0-08-055684-1 (e-book) OR 0123742889 (paperback), 0080556841 (ebook)
LCCN: QA47 .J38 2008
bibdate: Thu May 8 16:02:52 MDT 2008
bibtimestamp: 2008.05.08 16:02:52 MDT
subject: mathematics; tables; formulae
acknowledgement: ack-nhfb
rawauthor: Alan Jeffrey and Hui-Hui Dai
rawtitle: Handbook of Mathematical Formulas and Integrals
ISBN13:
978-0-12-374288-9 (paperback), 978-0-08-055684-0 (e-book) OR 9780123742889 (paperback), 9780080556840 (ebook)

bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/master.bib; https://www.math.utah.edu/pub/tex/bib/mathcw.bib; melvyl.cdlib.org:210/CDL90

URL:
https://shop.elsevier.com/books/handbook-of-mathematical-formulas-and-integrals/jeffrey/978-0-12-374288-9

TOC:
Preface Preface to the Fourth Edition Notes for Handbook Users Index of Special Functions and Notations 0. Quick Reference List of Frequently Used Data 1. Numerical, Algebraic, and Analytical Results for Series and Calculus 2. Functions and Identities 3. Derivatives of Elementary Functions 4. Indefinite Integrals of Algebraic Functions 5. Indefinite Integrals of Exponential Functions 6. Indefinite Integrals of Logarithmic Functions 7. Indefinite Integrals of Hyperbolic Functions 8. Indefinite Integrals Involving Inverse Hyperbolic Functions 9. Indefinite Integrals of Trigonometric Functions 10. Indefinite Integrals of Inverse Trigonometric Functions 11. The Gamma, Beta, Pi, and Psi Functions, and the Incomplete Gamma Functions 12. Elliptic Integrals and Functions 13. Probability Distributions and Integrals, and the Error Function 14. Fresnel Integrals, Sine and Cosine Integrals 15. Definite Integrals 16. Different Forms of Fourier Series 17. Bessel Functions 18. Orthogonal Polynomials 19. Laplace Transformation 20. Fourier Transforms 21. Numerical Integration 22. Solutions of Standard Ordinary Differential Equations 23. Vector Analysis 24. Systems of Orthogonal Coordinates 25. Partial Differential Equations and Special Functions 26. Qualitative Properties of the Heat and Laplace Equation 27. Solutions of Elliptic, Parabolic, and Hyperbolic Equations 28. The z-Transform 29. Numerical Approximation 30. Conformal Mapping and Boundary Value Problems Short Classified Reference List Index

rawTOC:
Preface \\ Preface to the Fourth Edition \\ Notes for Handbook Users \\ Index of Special Functions and Notations \\ 0. Quick Reference List of Frequently Used Data \\ 1. Numerical, Algebraic, and Analytical Results for Series and Calculus \\ 2. Functions and Identities \\ 3. Derivatives of Elementary Functions \\ 4. Indefinite Integrals of Algebraic Functions \\ 5. Indefinite Integrals of Exponential Functions \\ 6. Indefinite Integrals of Logarithmic Functions \\ 7. Indefinite Integrals of Hyperbolic Functions \\ 8. Indefinite Integrals Involving Inverse Hyperbolic Functions \\ 9. Indefinite Integrals of Trigonometric Functions \\ 10. Indefinite Integrals of Inverse Trigonometric Functions \\ 11. The Gamma, Beta, Pi, and Psi Functions, and the Incomplete Gamma Functions \\ 12. Elliptic Integrals and Functions \\ 13. Probability Distributions and Integrals, and the Error Function \\ 14. Fresnel Integrals, Sine and Cosine Integrals \\ 15. Definite Integrals \\ 16. Different Forms of Fourier Series \\ 17. Bessel Functions \\ 18. Orthogonal Polynomials \\ 19. Laplace Transformation \\ 20. Fourier Transforms \\ 21. Numerical Integration \\ 22. Solutions of Standard Ordinary Differential Equations \\ 23. Vector Analysis \\ 24. Systems of Orthogonal Coordinates \\ 25. Partial Differential Equations and Special Functions \\ 26. Qualitative Properties of the Heat and Laplace Equation \\ 27. Solutions of Elliptic, Parabolic, and Hyperbolic Equations \\ 28. The z-Transform \\ 29. Numerical Approximation \\ 30. Conformal Mapping and Boundary Value Problems \\ Short Classified Reference List \\ Index

entry:
@Book{Jeffrey:2008:HMF,
  author =       "Alan Jeffrey and Hui-Hui Dai",
  title =        "Handbook of Mathematical Formulas and Integrals",
  publisher =    pub-ELSEVIER-ACADEMIC,
  address =      pub-ELSEVIER-ACADEMIC:adr,
  edition =      "Fourth",
  pages =        "xlv + 541",
  year =         "2008",
  ISBN =         "0-12-374288-9 (paperback), 0-08-055684-1 (e-book)",
  ISBN-13 =      "978-0-12-374288-9 (paperback), 978-0-08-055684-0
                 (e-book)",
  LCCN =         "QA47 .J38 2008",
  bibdate =      "Thu May 8 16:02:52 MDT 2008",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/master.bib;
                 https://www.math.utah.edu/pub/tex/bib/mathcw.bib;
                 melvyl.cdlib.org:210/CDL90",
  URL =          "https://shop.elsevier.com/books/handbook-of-mathematical-formulas-and-integrals/jeffrey/978-0-12-374288-9",
  acknowledgement = ack-nhfb,
  subject =      "mathematics; tables; formulae",
  tableofcontents = "Preface \\
                 Preface to the Fourth Edition \\
                 Notes for Handbook Users \\
                 Index of Special Functions and Notations \\
                 0. Quick Reference List of Frequently Used Data \\
                 1. Numerical, Algebraic, and Analytical Results for
                 Series and Calculus \\
                 2. Functions and Identities \\
                 3. Derivatives of Elementary Functions \\
                 4. Indefinite Integrals of Algebraic Functions \\
                 5. Indefinite Integrals of Exponential Functions \\
                 6. Indefinite Integrals of Logarithmic Functions \\
                 7. Indefinite Integrals of Hyperbolic Functions \\
                 8. Indefinite Integrals Involving Inverse Hyperbolic
                 Functions \\
                 9. Indefinite Integrals of Trigonometric Functions \\
                 10. Indefinite Integrals of Inverse Trigonometric
                 Functions \\
                 11. The Gamma, Beta, Pi, and Psi Functions, and the
                 Incomplete Gamma Functions \\
                 12. Elliptic Integrals and Functions \\
                 13. Probability Distributions and Integrals, and the
                 Error Function \\
                 14. Fresnel Integrals, Sine and Cosine Integrals \\
                 15. Definite Integrals \\
                 16. Different Forms of Fourier Series \\
                 17. Bessel Functions \\
                 18. Orthogonal Polynomials \\
                 19. Laplace Transformation \\
                 20. Fourier Transforms \\
                 21. Numerical Integration \\
                 22. Solutions of Standard Ordinary Differential
                 Equations \\
                 23. Vector Analysis \\
                 24. Systems of Orthogonal Coordinates \\
                 25. Partial Differential Equations and Special
                 Functions \\
                 26. Qualitative Properties of the Heat and Laplace
                 Equation \\
                 27. Solutions of Elliptic, Parabolic, and Hyperbolic
                 Equations \\
                 28. The z-Transform \\
                 29. Numerical Approximation \\
                 30. Conformal Mapping and Boundary Value Problems \\
                 Short Classified Reference List \\
                 Index",
}


bibtype: article
filename: elefunt.bib
label: Kiani:2008:AND
author: M. Kiani and J. Panaretos and S. Psarakis and M. Saleem
journal: J. Iran. Stat. Soc.
volume: 7
number: 1
pages: 57--72
year: 2008
bibdate: Sat Dec 16 16:56:03 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2017.12.16 16:56:03 ???
acknowledgement: ack-nhfb
rawauthor: M. Kiani and J. Panaretos and S. Psarakis and M. Saleem
title:
Approximations to the normal distribution function and an extended table for the mean range of the normal variables

rawtitle:
Approximations to the normal distribution function and an extended table for the mean range of the normal variables

entry:
@Article{Kiani:2008:AND,
  author =       "M. Kiani and J. Panaretos and S. Psarakis and M.
                 Saleem",
  title =        "Approximations to the normal distribution function and
                 an extended table for the mean range of the normal
                 variables",
  journal =      "J. Iran. Stat. Soc.",
  volume =       "7",
  number =       "1",
  pages =        "57--72",
  month =        "????",
  year =         "2008",
  DOI =          "",
  bibdate =      "Sat Dec 16 16:56:03 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: elefunt.bib
label: Kodama:2008:ASP
author: Masao Kodama
journal: j-TOMS
volume: 34
number: 4
pages: 22:1--22:21
month: jul
monthnumber: 7
year: 2008
CODEN: ACMSCU
DOI: https://doi.org/10.1145/1377596.1377602
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
bibdate: Wed Jul 16 11:30:01 MDT 2008
bibtimestamp: 2008.07.16 11:30:01 MDT
fjournal: ACM Transactions on Mathematical Software (TOMS)
acknowledgement: ack-nhfb
articleno: 22
journalURL: https://dl.acm.org/loi/toms
rawauthor: Masao Kodama
title:
Algorithm 877: a Subroutine Package for Cylindrical Functions of Complex Order and Nonnegative Argument

abstract:
The algorithm presented provides a package of subroutines for calculating the cylindrical functions $ J_nu (x) $, $ N_nu (x) $, $ H_nu ^1 (x) $, $ H_nu ^2 (x) $ where the order $ nu $ is complex and the real argument $x$ is nonnegative. The algorithm is written in Fortran 95 and calculates the functions using single, double, or quadruple precision according to the value of a parameter defined in the algorithm. The methods of calculating the functions are based on a series expansion, Debye's asymptotic expansions, Olver's asymptotic expansions, and recurrence methods (Miller's algorithms). The relative errors of the functional values computed by this algorithm using double precision are less than $ 2.4 times 10^-13 $ in the region $ 0 leq mbox Re(nu ) leq 64 $, $ 0 leq mbox Im(nu ) leq 63 $, $ 0.024 leq x leq 97 $.

keywords:
Bessel functions; complex order; Cylindrical functions; Debye's asymptotic expansions; Hankel functions; Miller's algorithms; Neumann functions; nonnegative argument; numerical calculation; Olver's asymptotic expansions

rawabstract:
The algorithm presented provides a package of subroutines for calculating the cylindrical functions $ J_\nu (x) $, $ N_\nu (x) $, $ H_\nu^1 (x) $, $ H_\nu^2 (x) $ where the order $ \nu $ is complex and the real argument $x$ is nonnegative. The algorithm is written in Fortran 95 and calculates the functions using single, double, or quadruple precision according to the value of a parameter defined in the algorithm. The methods of calculating the functions are based on a series expansion, Debye's asymptotic expansions, Olver's asymptotic expansions, and recurrence methods (Miller's algorithms). The relative errors of the functional values computed by this algorithm using double precision are less than $ 2.4 \times 10^{-13} $ in the region $ 0 \leq \mbox {Re}(\nu) \leq 64 $, $ 0 \leq \mbox {Im}(\nu) \leq 63 $, $ 0.024 \leq x \leq 97 $.

rawtitle:
{Algorithm 877}: a Subroutine Package for Cylindrical Functions of Complex Order and Nonnegative Argument

entry:
@Article{Kodama:2008:ASP,
  author =       "Masao Kodama",
  title =        "{Algorithm 877}: a Subroutine Package for Cylindrical
                 Functions of Complex Order and Nonnegative Argument",
  journal =      j-TOMS,
  volume =       "34",
  number =       "4",
  pages =        "22:1--22:21",
  month =        jul,
  year =         "2008",
  CODEN =        "ACMSCU",
  DOI =          "https://doi.org/10.1145/1377596.1377602",
  ISSN =         "0098-3500 (print), 1557-7295 (electronic)",
  ISSN-L =       "0098-3500",
  bibdate =      "Wed Jul 16 11:30:01 MDT 2008",
  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 =     "The algorithm presented provides a package of
                 subroutines for calculating the cylindrical functions $
                 J_\nu (x) $, $ N_\nu (x) $, $ H_\nu^1 (x) $, $ H_\nu^2
                 (x) $ where the order $ \nu $ is complex and the real
                 argument $x$ is nonnegative. The algorithm is written
                 in Fortran 95 and calculates the functions using
                 single, double, or quadruple precision according to the
                 value of a parameter defined in the algorithm. The
                 methods of calculating the functions are based on a
                 series expansion, Debye's asymptotic expansions,
                 Olver's asymptotic expansions, and recurrence methods
                 (Miller's algorithms). The relative errors of the
                 functional values computed by this algorithm using
                 double precision are less than $ 2.4 \times 10^{-13} $
                 in the region $ 0 \leq \mbox {Re}(\nu) \leq 64 $, $ 0
                 \leq \mbox {Im}(\nu) \leq 63 $, $ 0.024 \leq x \leq 97
                 $.",
  acknowledgement = ack-nhfb,
  articleno =    "22",
  fjournal =     "ACM Transactions on Mathematical Software (TOMS)",
  journal-URL =  "https://dl.acm.org/loi/toms",
  keywords =     "Bessel functions; complex order; Cylindrical
                 functions; Debye's asymptotic expansions; Hankel
                 functions; Miller's algorithms; Neumann functions;
                 nonnegative argument; numerical calculation; Olver's
                 asymptotic expansions",
}


bibtype: article
filename: elefunt.bib
label: Lefevre:2008:WCE
author: Vincent Lefevre and Damien Stehle and Paul Zimmermann
title: Worst Cases for the Exponential Function in the IEEE 754r decimal64 Format
journal: j-LECT-NOTES-COMP-SCI
volume: 5045
pages: 114--126
year: 2008
CODEN: LNCSD9
DOI: https://doi.org/10.1007/978-3-540-85521-7_7
ISSN: 0302-9743 (print), 1611-3349 (electronic) OR 03029743 (print), 16113349 (electronic)
ISSNL: 0302-9743 OR 03029743
bibdate: Thu Oct 1 11:29:36 MDT 2015
bibtimestamp: 2015.10.01 11:29:36 MDT
URL: http://link.springer.com/content/pdf/10.1007/978-3-540-85521-7_7.pdf
fjournal: Lecture Notes in Computer Science
acknowledgement: ack-nhfb
bookDOI: https://doi.org/10.1007/978-3-540-85521-7
bookURL: http://www.springerlink.com/content/978-3-540-85521-7
journalURL: http://link.springer.com/bookseries/558
rawauthor: Vincent Lef{\`e}vre and Damien Stehl{\'e} and Paul Zimmermann
rawtitle: Worst Cases for the Exponential Function in the {IEEE 754r decimal64} Format
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/lncs2008a.bib

remark:
From the abstract: ``the worst case for $ |x| geq 3 times 10^-11 $ is exp(9.407822313572878e-2) = 1.09864568206633850000000000000000278.''

entry:
@Article{Lefevre:2008:WCE,
  author =       "Vincent Lef{\`e}vre and Damien Stehl{\'e} and Paul
                 Zimmermann",
  title =        "Worst Cases for the Exponential Function in the {IEEE
                 754r decimal64} Format",
  journal =      j-LECT-NOTES-COMP-SCI,
  volume =       "5045",
  pages =        "114--126",
  year =         "2008",
  CODEN =        "LNCSD9",
  DOI =          "https://doi.org/10.1007/978-3-540-85521-7_7",
  ISSN =         "0302-9743 (print), 1611-3349 (electronic)",
  ISSN-L =       "0302-9743",
  bibdate =      "Thu Oct 1 11:29:36 MDT 2015",
  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/lncs2008a.bib",
  URL =          "http://link.springer.com/content/pdf/10.1007/978-3-540-85521-7_7.pdf",
  acknowledgement = ack-nhfb,
  book-DOI =     "https://doi.org/10.1007/978-3-540-85521-7",
  book-URL =     "http://www.springerlink.com/content/978-3-540-85521-7",
  fjournal =     "Lecture Notes in Computer Science",
  journal-URL =  "http://link.springer.com/bookseries/558",
  remark =       "From the abstract: ``the worst case for $ |x| \geq 3
                 \times 10^{-11} $ is exp(9.407822313572878e-2) =
                 1.09864568206633850000000000000000278.''",
}


bibtype: article
filename: elefunt.bib
label: Lorch:2008:MSR
author: Lee Lorch and Martin E. Muldoon
title: Monotonic sequences related to zeros of Bessel functions
journal: j-NUMER-ALGORITHMS
volume: 49
number: 1--4
pages: 221--233
month: dec
monthnumber: 12
year: 2008
CODEN: NUALEG
ISSN: 1017-1398 (print), 1572-9265 (electronic) OR 10171398 (print), 15729265 (electronic)
ISSNL: 1017-1398 OR 10171398
bibdate: Mon May 17 15:07:19 MDT 2010
bibtimestamp: 2010.05.17 15:07:19 MDT
URL: http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=49&issue=1&spage=221
fjournal: Numerical Algorithms
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11075
rawauthor: Lee Lorch and Martin E. Muldoon
rawtitle: Monotonic sequences related to zeros of {Bessel} functions
entry:
@Article{Lorch:2008:MSR,
  author =       "Lee Lorch and Martin E. Muldoon",
  title =        "Monotonic sequences related to zeros of {Bessel}
                 functions",
  journal =      j-NUMER-ALGORITHMS,
  volume =       "49",
  number =       "1--4",
  pages =        "221--233",
  month =        dec,
  year =         "2008",
  CODEN =        "NUALEG",
  ISSN =         "1017-1398 (print), 1572-9265 (electronic)",
  ISSN-L =       "1017-1398",
  bibdate =      "Mon May 17 15:07:19 MDT 2010",
  bibsource =    "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1017-1398&volume=49&issue=1;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=49&issue=1&spage=221",
  acknowledgement = ack-nhfb,
  fjournal =     "Numerical Algorithms",
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bibtype: book
filename: elefunt.bib
label: Mathai:2008:SFA
author: A. M. Mathai and H. J. Haubold
title: Special Functions for Applied Scientists
publisher: Springer Science+Business Media
address: New York, NY, USA
pages: xxv + 464
year: 2008
DOI: https://doi.org/10.1007/978-0-387-75894-7
ISBN: 0-387-75893-3 OR 0387758933
ISBN13: 978-0-387-75893-0 OR 9780387758930
LCCN: QA351 .M37X 2008
bibdate: Sat Oct 30 17:02:02 MDT 2010
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib; prodorbis.library.yale.edu:7090/voyager
bibtimestamp: 2010.10.30 17:02:02 MDT
subject: Functions, Special; Fractional calculus; Wavelets (Mathematics)
acknowledgement: ack-nhfb
rawauthor: A. M. Mathai and H. J. Haubold
rawtitle: Special Functions for Applied Scientists
TOC:
1 Basic Ideas of Special Functions and Statistical Distributions / 1 1.0 Introduction / 1 1.1 Gamma Function / 3 1.1.1 Some basic properties of gamma functions / 4 1.1.2 Wedge product and Jacobians of transformations / 6 1.1.3 Multiplication formula for gamma functions / 8 1.1.4 Asymptotic formula for a gamma function / 9 1.1.5 Bernoulli polynomials / 9 1.1.6 Some basic properties of generalized Bernoulli polynomials / 9 1.1.7 The first three generalized Bernoulli polynomials / 10 Exercises 1.1 / 11 1.2 The Psi and Zeta Functions / 12 1.2.1 Generalized zeta function / 12 Exercises 1.2 / 13 1.3 Integral Transforms / 14 1.3.1 Mellin transform / 14 1.3.2 Laplace transform / 17 Exercises 1.3 / 18 1.4 Some Statistical Preliminaries / 19 Exercises 1.4 / 26 1.5 Some Properties of Random Variables / 28 1.5.1 Multivariate analogues / 30 1.5.2 Marginal and conditional densities / 31 Exercises 1.5 / 33 1.6 Beta and Related Functions / 34 1.6.1 Dirichlet integrals and Dirichlet densities / 37 Exercises 1.6 / 42 1.7 Hypergeometric Series / 42 1.7.1 Evaluation of some contour integrals / 45 1.7.2 Residues when several gammas are involved / 46 Exercises 1.7 / 48 1.8 Meijer's $G$-function / 49 Exercises 1.8 / 54 1.9 The $H$-function / 54 Exercises 1.9 / 57 1.10 Lauricella Functions and Appell's Functions / 58 1.10.1 Some properties of $f_A$ / 59 1.10.2 Some properties of $f_B$ / 60 1.10.3 Some properties of $f_C$ / 60 1.10.4 Some properties of $f_D$ / 61 Exercises 1.10 / 64 1.11 Special Functions as Solutions of Differential Equations and Applications / 64 1.11.0 Introduction / 64 1.11.1 Sine, cosine and exponential functions / 64 Exercises 1.11 / 66 1.11.2 Linear second order differential equations / 66 1.11.3 Hypergeometric function / 67 Exercises 1.11 / 69 1.11.4 Confluent hypergeometric function / 70 Exercises 1.11 / 71 1.11.5 Hermite polynomials / 71 Exercises 1.11 / 72 1.11.6 Bessel functions / 72 Exercises 1.11 / 73 1.11.7 Laguerre polynomial / 74 1.11.8 Legendre polynomial / 74 Exercises 1.11 / 74 1.11.9 Generalized hypergeometric function / 75 1.11.10 $G$-function / 76 Exercises 1.11 / 77 References / 77 2 Mittag-Leffler Functions and Fractional Calculus / 79 2.0 Introduction / 79 2.1 Mittag-Leffler Function / 79 Revision Exercises 2.1 / 81 2.2 Basic Properties of Mittag-Leffler Function / 82 2.2.1 Mittag-Leffler functions of rational order / 84 2.2.2 Euler transform of Mittag-Leffler function / 84 2.2.3 Laplace transform of Mittag-Leffler function / 85 2.2.4 Application of Laplace transform / 87 2.2.5 Mittag-Leffler functions and the $H$-function / 88 Exercises 2.2 / 90 2.3 Generalized Mittag-Leffler Function / 91 2.3.1 Special cases of $E_beta , gamma ^delta (z)$ / 92 2.3.2 Mellin--Barnes integral representation / 92 2.3.3 Relations with the H-function and Wright function / 92 2.3.4 Cases of reducibility / 93 2.3.5 Differentiation of generalized Mittag-Leffler function / 94 2.3.6 Integral property of generalized Mittag-Leffler function / 94 2.3.7 Integral transform of $E_beta , gamma ^delta (z)$ / 95 Exercises 2.3 / 96 2.4 Fractional Integrals / 97 2.4.1 Riemann--Liouville fractional integrals of arbitrary order / 98 2.4.2 Riemann--Liouville fractional integrals of order $alpha $ / 99 2.4.3 Basic properties of fractional integrals / 100 2.4.4 A useful integral / 101 2.4.5 The Weyl integral / 103 2.4.6 Basic properties of Weyl integral / 104 Exercises 2.4 / 105 2.4.7 Laplace transform of the fractional integral / 107 2.4.8 Laplace transform of the fractional derivative / 107 2.4.9 Laplace transform of Caputo derivative / 108 Exercises 2.4 / 109 2.5 Mellin Transform of the Fractional Integrals and the Fractional Derivatives / 109 2.5.1 Mellin transform / 109 2.5.2 Mellin transform of the fractional integral / 110 2.5.3 Mellin transform of the fractional derivative / 111 Exercises 2.5 / 111 2.6 Kober Operators / 111 Exercises 2.6 / 113 2.7 Generalized Kober Operators / 114 Exercises 2.7 / 118 2.8 Compositions of Riemann--Liouville Fractional Calculus Operators and Generalized Mittag-Leffler Functions / 121 2.8.1 Composition relations between R-L operators and $E_beta , gamma ^delta (z)$ / 121 Exercises 2.8 / 126 2.9 Fractional Differential Equations / 126 2.9.1 Fractional relaxation / 127 Exercises 2.9 / 130 2.9.2 Fractional diffusion / 131 Exercises 2.9 / 132 References / 133 3 An Introduction to $q$-Series / 135 3.0 Introduction / 135 3.1 Hypergeometric Series / 135 Exercises 3.1 / 136 3.2 Basic Hypergeometric Series ($q$-Series) / 138 Exercises 3.2 / 140 3.2.1 The $q$-binomial theorem / 142 Exercises 3.2 / 146 3.2.2 The $q$-binomial coefficients / 146 Exercises 3.2 / 148 3.3 $q$-Calculus / 149 Exercises 3.3 / 151 3.4 The $q$-Gamma and $q$-Beta Functions / 151 3.5 Transformation and Summation Formulas for $q$-Series / 152 Exercises 3.5 / 154 3.6 Jacobi's Triple Product and Rogers--Ramanujan Identities / 154 References / 157 4 Ramanujan's Theories of Theta and Elliptic Functions / 159 4.0 Introduction / 159 4.1 Ramanujan's Theory of Classical Theta Functions / 159 4.1.1 Series definition and additive results / 159 4.2 Ramanujan's $_1Psi _1$ Summation Formula and Multiplicative Results for Theta Functions / 163 4.3 Modular Equations / 170 4.4 Inversion Formulas and Evaluations / 172 4.5 Modular Identities (Classical Theory) / 175 4.6 Ramanujan's Theory of Cubic Theta Functions / 177 4.6.1 The cubic theta functions / 177 4.6.2 Inversion formulas and evaluations (cubic theory) / 179 4.6.3 Triplication and trimediation formulas / 182 4.6.4 Further evaluations / 183 4.6.5 Evaluations of Ramanujan--Eisenstein series ($L$, $M$, $N$ or $P$, $Q$, $R$) / 185 4.6.6 The cubic analogue of the Jacobian elliptic functions / 186 Test on Ramanujan's work / 187 4.7 The One-variable Cubic Theta Functions / 188 4.7.1 Cubic theta functions and some properties / 189 4.7.2 Product representations for $b(q)$ and $c(q)$ / 190 4.7.3 The cubic analogue of Jacobi's quartic modular equations / 191 4.8 The Two-variable Cubic Theta Functions / 193 4.8.1 Series definitions and some properties / 193 4.8.2 Product representations for $b(q,z)$ and $c(q,z)$ / 195 4.8.3 A two-variable cubic counterpart of Jacobi's quartic modular equation / 200 4.9 The Three-variable Cubic Theta Functions / 200 4.9.1 Unification of one and two-variable cubic theta functions / 200 Exercises 4.9 / 200 4.9.2 Generalization of Hirschhorn--Garvan--Borwein identity / 202 4.9.3 Laurent's expansions for two-parameter cubic theta functions / 205 References / 209 5 Lie Group and Special Functions / 211 5.1 General Introduction to Group Theory / 211 5.1.1 Isomorphisms / 213 5.1.2 Symmetry groups / 213 5.1.3 Isometries of the Euclidean plane / 214 5.1.4 Finite groups of motion / 215 5.1.5 Discrete groups of motions / 216 5.2 Lie Group and Special Functions / 217 5.2.1 Subspace of a vector space / 218 5.3 Lie Algebra / 219 5.4 Representations of Lie Algebra / 222 Exercises 5.4 / 225 5.5 Special Functions / 226 5.5.1 Gauss hypergeometric function / 226 5.5.2 Differential equation satisfied by $_2F_1$ / 228 5.5.3 Integral representation of $_F_q(alpha _1,alpha _2, ldots , alpha _p beta _1, beta _2, ldots , beta _p; z)$ / 230 5.6 Laguerre Polynomial $L^(alpha )_n(x)$ / 231 5.6.1 Laguerre polynomial and Lie algebra / 232 Exercises 5.6 / 232 5.7 Helmholtz Equation / 234 5.7.1 Helmholtz equation in three variables / 238 5.8 Lie Group / 241 5.8.1 Basis of the Lie algebra of the Lie group SL(2) / 242 Exercises 5.8 / 243 Test on Lie Theory and Special Functions / 244 6 Applications to Stochastic Process and Time Series / 247 6.0 Introduction / 247 6.1 Stochastic Processes / 247 6.1.1 Classical types of stochastic processes / 252 6.1.2 Processes with stationary independent increments / 252 6.1.3 Stationary processes / 253 6.1.4 Gaussian processes and stationarity / 253 6.1.5 Brownian processes / 257 6.1.6 Markov chains / 258 Exercises 6.1 / 261 6.2 Modern Concepts in Distribution Theory / 262 6.2.1 Introduction / 262 6.2.2 Geometric infinite divisibility / 263 6.2.3 Bernstein functions / 264 6.2.4 Self-decomposability / 265 6.2.5 Stable distributions / 265 6.2.6 Geometrically strictly stable distributions / 266 6.2.7 Mittag-Leffler distribution / 266 6.2.8 $alpha $-Laplace distribution / 267 6.2.9 Semi-Pareto distribution / 268 Exercises 6.2 / 268 6.3 Stationary Time Series Models / 269 6.3.1 Introduction / 269 6.3.2 Autoregressive models / 270 6.3.3 A general solution / 270 6.3.4 Extension to a $k$-th order autoregressive model / 272 6.3.5 Mittag-Leffler autoregressive structure / 273 Exercises 6.3 / 274 6.4 A Structural Relationship and New Processes / 275 6.4.1 The TMLAR(1) process / 276 6.4.2 The NEAR(1) model / 277 6.4.3 New Mittag-Leffler autoregressive models / 278 6.4.4 The NSMLAR(1) process / 280 Exercises 6.4 / 281 6.5 Tailed Processes / 281 6.5.1 The exponential tailed autoregressive process [ETAR(1)] / 282 6.5.2 The Mittag-Leffler tailed autoregressive process [MLTAR(1)] / 283 Exercises 6.5 / 286 6.6 Marshall--Olkin Weibull Time Series Models / 286 6.6.1 Introduction / 286 6.6.2 Marshall--Olkin semi-Weibull distribution / 287 6.6.3 An AR(1) model with MOSW marginal distribution / 289 6.6.4 Marshall--Olkin generalized Weibull distribution / 291 6.6.5 An AR(1) model with MOGW marginal distribution / 292 6.6.6 Case study / 293 Exercises 6.6 / 293 References / 294 7 Applications to Density Estimation / 297 7.0 Density Estimation and Orthogonal Polynomials / 297 7.1 Introduction / 297 7.2 Approximants Based on Legendre Polynomials / 299 7.3 Approximants Based on Laguerre Polynomials / 301 7.4 A Unified Methodology / 304 7.5 Approximants Expressed in Terms of Orthogonal Polynomials / 305 7.5.1 Approximants expressed in terms of Laguerre polynomials / 306 7.5.2 Approximants expressed in terms of Legendre polynomials / 306 7.5.3 Approximants expressed in terms of Jacobi polynomials / 307 7.5.4 Approximants expressed in terms of Hermite polynomials / 307 References / 308 8 Applications to Order Statistics / 311 8.0 Introduction / 311 8.1 Distribution Function / 311 8.1.1 Density of the $r$-th order statistic / 313 8.1.2 Joint distribution function of two order statistics / 314 8.1.3 Joint density of two order statistics / 314 8.1.4 Moments of order statistics / 315 8.1.5 Recurrence relations for moments / 317 8.1.6 Recurrence relations on the product moments / 318 8.1.7 Order statistics from symmetric distributions / 319 8.2 Discrete Order Statistics / 319 8.2.1 Probability function of discrete order statistics / 320 8.2.2 Joint probability function of two order statistics / 321 8.2.3 Bernoulli order statistics / 321 8.3 Independent Random Variables / 322 8.3.1 Distribution of a single order statistic / 322 8.3.2 Joint distribution of two order statistics / 324 Test on Order Statistics / 325 8.4 On Concomitants of Order Statistics / 326 8.4.1 Application of concomitants of order statistics / 326 8.4.2 Application in estimation / 328 8.4.3 Concomitants of record values and estimation problems / 333 References / 339 9 Applications to Astrophysics Problems / 341 9.0 Introduction / 341 9.1 Entropy: Boltzmann, Planck, and Einstein on W / 342 9.1.1 Entropic functional / 342 9.1.2 Entropy and probability / 342 9.1.3 Boltzmann--Gibbs / 343 9.2 Gravitationally Stabilized Fusion Reactor: The Sun / 345 9.2.1 Internal solar structure / 345 9.2.2 Solar fusion plasma / 349 9.2.3 Estimation of central temperature in the Sun / 349 9.3 Crucial Astrophysical Experiments: Data Analysis / 351 9.3.1 The experiments / 351 9.3.2 Analysis of the time series / 352 9.4 Fundamental Equations for Nonequilibrium Processes / 355 9.4.1 Chapman--Kolmogorov equation / 355 9.4.2 Master equation / 355 9.4.3 Fokker--Planck equation / 356 9.4.4 Langevin equation / 357 9.4.5 Reaction-diffusion equation / 357 9.5 Fractional Calculus / 360 9.6 Nonextensive Statistical Mechanics / 362 9.7 Standard and Fractional Reaction / 363 9.7.1 Boltzmann--Gibbs statistical mechanics / 363 9.7.2 Generalized Boltzmann--Gibbs statistical mechanics / 366 9.7.3 Fractional reaction / 369 9.7.4 Thermonuclear reaction coefficient / 371 9.8 Standard and Fractional Diffusion / 380 9.8.1 Fick's first law of diffusion / 380 9.8.2 Einstein's approach to diffusion / 381 9.8.3 Fractional diffusion / 381 9.8.4 Spatio-temporal pattern formation / 383 References / 384 10 An Introduction to Wavelet Analysis / 389 10.0 Introduction / 389 10.1 Fourier Analysis to Wavelet Analysis / 390 10.2 Construction of Orthonormal Wavelets / 393 10.3 Classification of Wavelets and Multiresolution Analysis / 400 10.4 Spline Wavelets / 405 10.5 A Variant of Construction of Orthonormal Wavelets / 407 Exercises 10.5 / 408 References / 408 11 Jacobians of Matrix Transformations / 409 11.0 Introduction / 409 11.1 Jacobians of Linear Matrix Transformations / 410 Exercises 11.1 / 415 11.2 Jacobians in Some Nonlinear Transformations / 417 Exercises 11.2 / 421 11.3 Transformations Involving Orthonormal Matrices / 423 Exercises 11.3 / 426 References / 428 12 Special Functions of Matrix Argument / 429 12.0 Introduction / 429 12.1 Real Matrix-Variate Scalar Functions / 429 12.1.1 Real matrix-variate gamma / 430 12.1.2 Real matrix-variate gamma density / 430 Exercises 12.1 / 434 12.2 The Laplace Transform in the Matrix Case / 435 12.2.1 A convolution property for Laplace transforms / 436 Exercises 12.2 / 439 12.3 Hypergeometric Functions with Matrix Argument / 439 12.3.1 Hypergeometric function through Laplace transform / 440 12.3.2 Hypergeometric function through zonal polynomials / 441 12.3.3 Hypergeometric functions through $M$-transforms / 443 12.3.4 A convolution theorem for $M$-transforms / 445 Exercises 12.3 / 446 12.4 A Pathway Model / 448 12.4.1 The pathway density / 448 12.4.2 A general density / 451 12.4.3 Arbitrary moments / 451 12.4.4 Quadratic forms / 452 12.4.5 Generalized quadratic form / 452 12.4.6 Applications to random volumes / 453 Exercises 12.4 / 453 References / 454 Author Index / 457 Subject Index / 461

shortTOC:
Basic ideas of special functions and statistical distributions Mittag-Leffler functions and fractional calculus An introduction to q-series Ramanujan's theories of theta and elliptic functions Lie group and special functions Applications to stochastic process and time series Applications to density estimation Applications to order statistics Applications to astrophysics problems An introduction to wavelet analysis Jacobians of matrix formations Special functions of matrix argument

rawshortTOC:
Basic ideas of special functions and statistical distributions \\ Mittag-Leffler functions and fractional calculus \\ An introduction to q-series \\ Ramanujan's theories of theta and elliptic functions \\ Lie group and special functions \\ Applications to stochastic process and time series \\ Applications to density estimation \\ Applications to order statistics \\ Applications to astrophysics problems \\ An introduction to wavelet analysis \\ Jacobians of matrix formations \\ Special functions of matrix argument

rawTOC:
1 Basic Ideas of Special Functions and Statistical Distributions / 1 \\ 1.0 Introduction / 1 \\ 1.1 Gamma Function / 3 \\ 1.1.1 Some basic properties of gamma functions / 4 \\ 1.1.2 Wedge product and Jacobians of transformations / 6 \\ 1.1.3 Multiplication formula for gamma functions / 8 \\ 1.1.4 Asymptotic formula for a gamma function / 9 \\ 1.1.5 Bernoulli polynomials / 9 \\ 1.1.6 Some basic properties of generalized Bernoulli polynomials / 9 \\ 1.1.7 The first three generalized Bernoulli polynomials / 10 \\ Exercises 1.1 / 11 \\ 1.2 The Psi and Zeta Functions / 12 \\ 1.2.1 Generalized zeta function / 12 \\ Exercises 1.2 / 13 \\ 1.3 Integral Transforms / 14 \\ 1.3.1 Mellin transform / 14 \\ 1.3.2 Laplace transform / 17 \\ Exercises 1.3 / 18 \\ 1.4 Some Statistical Preliminaries / 19 \\ Exercises 1.4 / 26 \\ 1.5 Some Properties of Random Variables / 28 \\ 1.5.1 Multivariate analogues / 30 \\ 1.5.2 Marginal and conditional densities / 31 \\ Exercises 1.5 / 33 \\ 1.6 Beta and Related Functions / 34 \\ 1.6.1 Dirichlet integrals and Dirichlet densities / 37 \\ Exercises 1.6 / 42 \\ 1.7 Hypergeometric Series / 42 \\ 1.7.1 Evaluation of some contour integrals / 45 \\ 1.7.2 Residues when several gammas are involved / 46 \\ Exercises 1.7 / 48 \\ 1.8 Meijer's $G$-function / 49 \\ Exercises 1.8 / 54 \\ 1.9 The $H$-function / 54 \\ Exercises 1.9 / 57 \\ 1.10 Lauricella Functions and Appell's Functions / 58 \\ 1.10.1 Some properties of $f_A$ / 59 \\ 1.10.2 Some properties of $f_B$ / 60 \\ 1.10.3 Some properties of $f_C$ / 60 \\ 1.10.4 Some properties of $f_D$ / 61 \\ Exercises 1.10 / 64 \\ 1.11 Special Functions as Solutions of Differential Equations and Applications / 64 \\ 1.11.0 Introduction / 64 \\ 1.11.1 Sine, cosine and exponential functions / 64 \\ Exercises 1.11 / 66 \\ 1.11.2 Linear second order differential equations / 66 \\ 1.11.3 Hypergeometric function / 67 \\ Exercises 1.11 / 69 \\ 1.11.4 Confluent hypergeometric function / 70 \\ Exercises 1.11 / 71 \\ 1.11.5 Hermite polynomials / 71 \\ Exercises 1.11 / 72 \\ 1.11.6 Bessel functions / 72 \\ Exercises 1.11 / 73 \\ 1.11.7 Laguerre polynomial / 74 \\ 1.11.8 Legendre polynomial / 74 \\ Exercises 1.11 / 74 \\ 1.11.9 Generalized hypergeometric function / 75 \\ 1.11.10 $G$-function / 76 \\ Exercises 1.11 / 77 \\ References / 77 \\ 2 Mittag-Leffler Functions and Fractional Calculus / 79 \\ 2.0 Introduction / 79 \\ 2.1 Mittag-Leffler Function / 79 \\ Revision Exercises 2.1 / 81 \\ 2.2 Basic Properties of Mittag-Leffler Function / 82 \\ 2.2.1 Mittag-Leffler functions of rational order / 84 \\ 2.2.2 Euler transform of Mittag-Leffler function / 84 \\ 2.2.3 Laplace transform of Mittag-Leffler function / 85 \\ 2.2.4 Application of Laplace transform / 87 \\ 2.2.5 Mittag-Leffler functions and the $H$-function / 88 \\ Exercises 2.2 / 90 \\ 2.3 Generalized Mittag-Leffler Function / 91 \\ 2.3.1 Special cases of $E_{\beta, \gamma}^\delta (z)$ / 92 \\ 2.3.2 Mellin--Barnes integral representation / 92 \\ 2.3.3 Relations with the H-function and Wright function / 92 \\ 2.3.4 Cases of reducibility / 93 \\ 2.3.5 Differentiation of generalized Mittag-Leffler function / 94 \\ 2.3.6 Integral property of generalized Mittag-Leffler function / 94 \\ 2.3.7 Integral transform of $E_{\beta, \gamma}^\delta (z)$ / 95 \\ Exercises 2.3 / 96 \\ 2.4 Fractional Integrals / 97 \\ 2.4.1 Riemann--Liouville fractional integrals of arbitrary order / 98 \\ 2.4.2 Riemann--Liouville fractional integrals of order $\alpha$ / 99 \\ 2.4.3 Basic properties of fractional integrals / 100 \\ 2.4.4 A useful integral / 101 \\ 2.4.5 The Weyl integral / 103 \\ 2.4.6 Basic properties of Weyl integral / 104 \\ Exercises 2.4 / 105 \\ 2.4.7 Laplace transform of the fractional integral / 107 \\ 2.4.8 Laplace transform of the fractional derivative / 107 \\ 2.4.9 Laplace transform of Caputo derivative / 108 \\ Exercises 2.4 / 109 \\ 2.5 Mellin Transform of the Fractional Integrals and the Fractional Derivatives / 109 \\ 2.5.1 Mellin transform / 109 \\ 2.5.2 Mellin transform of the fractional integral / 110 \\ 2.5.3 Mellin transform of the fractional derivative / 111 \\ Exercises 2.5 / 111 \\ 2.6 Kober Operators / 111 \\ Exercises 2.6 / 113 \\ 2.7 Generalized Kober Operators / 114 \\ Exercises 2.7 / 118 \\ 2.8 Compositions of Riemann--Liouville Fractional Calculus Operators and Generalized Mittag-Leffler Functions / 121 \\ 2.8.1 Composition relations between R-L operators and $E_{\beta, \gamma}^\delta (z)$ / 121 \\ Exercises 2.8 / 126 \\ 2.9 Fractional Differential Equations / 126 \\ 2.9.1 Fractional relaxation / 127 \\ Exercises 2.9 / 130 \\ 2.9.2 Fractional diffusion / 131 \\ Exercises 2.9 / 132 \\ References / 133 \\ 3 An Introduction to $q$-Series / 135 \\ 3.0 Introduction / 135 \\ 3.1 Hypergeometric Series / 135 \\ Exercises 3.1 / 136 \\ 3.2 Basic Hypergeometric Series ($q$-Series) / 138 \\ Exercises 3.2 / 140 \\ 3.2.1 The $q$-binomial theorem / 142 \\ Exercises 3.2 / 146 \\ 3.2.2 The $q$-binomial coefficients / 146 \\ Exercises 3.2 / 148 \\ 3.3 $q$-Calculus / 149 \\ Exercises 3.3 / 151 \\ 3.4 The $q$-Gamma and $q$-Beta Functions / 151 \\ 3.5 Transformation and Summation Formulas for $q$-Series / 152 \\ Exercises 3.5 / 154 \\ 3.6 Jacobi's Triple Product and Rogers--Ramanujan Identities / 154 \\ References / 157 \\ 4 Ramanujan's Theories of Theta and Elliptic Functions / 159 \\ 4.0 Introduction / 159 \\ 4.1 Ramanujan's Theory of Classical Theta Functions / 159 \\ 4.1.1 Series definition and additive results / 159 \\ 4.2 Ramanujan's $_1\Psi_1$ Summation Formula and Multiplicative Results for Theta Functions / 163 \\ 4.3 Modular Equations / 170 \\ 4.4 Inversion Formulas and Evaluations / 172 \\ 4.5 Modular Identities (Classical Theory) / 175 \\ 4.6 Ramanujan's Theory of Cubic Theta Functions / 177 \\ 4.6.1 The cubic theta functions / 177 \\ 4.6.2 Inversion formulas and evaluations (cubic theory) / 179 \\ 4.6.3 Triplication and trimediation formulas / 182 \\ 4.6.4 Further evaluations / 183 \\ 4.6.5 Evaluations of Ramanujan--Eisenstein series ($L$, $M$, $N$ or $P$, $Q$, $R$) / 185 \\ 4.6.6 The cubic analogue of the Jacobian elliptic functions / 186 \\ Test on Ramanujan's work / 187 \\ 4.7 The One-variable Cubic Theta Functions / 188 \\ 4.7.1 Cubic theta functions and some properties / 189 \\ 4.7.2 Product representations for $b(q)$ and $c(q)$ / 190 \\ 4.7.3 The cubic analogue of Jacobi's quartic modular equations / 191 \\ 4.8 The Two-variable Cubic Theta Functions / 193 \\ 4.8.1 Series definitions and some properties / 193 \\ 4.8.2 Product representations for $b(q,z)$ and $c(q,z)$ / 195 \\ 4.8.3 A two-variable cubic counterpart of Jacobi's quartic modular equation / 200 \\ 4.9 The Three-variable Cubic Theta Functions / 200 \\ 4.9.1 Unification of one and two-variable cubic theta functions / 200 \\ Exercises 4.9 / 200 \\ 4.9.2 Generalization of Hirschhorn--Garvan--Borwein identity / 202 \\ 4.9.3 Laurent's expansions for two-parameter cubic theta functions / 205 \\ References / 209 \\ 5 Lie Group and Special Functions / 211 \\ 5.1 General Introduction to Group Theory / 211 \\ 5.1.1 Isomorphisms / 213 \\ 5.1.2 Symmetry groups / 213 \\ 5.1.3 Isometries of the Euclidean plane / 214 \\ 5.1.4 Finite groups of motion / 215 \\ 5.1.5 Discrete groups of motions / 216 \\ 5.2 Lie Group and Special Functions / 217 \\ 5.2.1 Subspace of a vector space / 218 \\ 5.3 Lie Algebra / 219 \\ 5.4 Representations of Lie Algebra / 222 \\ Exercises 5.4 / 225 \\ 5.5 Special Functions / 226 \\ 5.5.1 Gauss hypergeometric function / 226 \\ 5.5.2 Differential equation satisfied by $_2F_1$ / 228 \\ 5.5.3 Integral representation of $_F_q(\alpha_1,\alpha_2, \ldots{}, \alpha_p \beta_1, \beta_2, \ldots{}, \beta_p; z)$ / 230 \\ 5.6 Laguerre Polynomial $L^{(\alpha)}_n(x)$ / 231 \\ 5.6.1 Laguerre polynomial and Lie algebra / 232 \\ Exercises 5.6 / 232 \\ 5.7 Helmholtz Equation / 234 \\ 5.7.1 Helmholtz equation in three variables / 238 \\ 5.8 Lie Group / 241 \\ 5.8.1 Basis of the Lie algebra of the Lie group SL(2) / 242 \\ Exercises 5.8 / 243 \\ Test on Lie Theory and Special Functions / 244 \\ 6 Applications to Stochastic Process and Time Series / 247 \\ 6.0 Introduction / 247 \\ 6.1 Stochastic Processes / 247 \\ 6.1.1 Classical types of stochastic processes / 252 \\ 6.1.2 Processes with stationary independent increments / 252 \\ 6.1.3 Stationary processes / 253 \\ 6.1.4 Gaussian processes and stationarity / 253 \\ 6.1.5 Brownian processes / 257 \\ 6.1.6 Markov chains / 258 \\ Exercises 6.1 / 261 \\ 6.2 Modern Concepts in Distribution Theory / 262 \\ 6.2.1 Introduction / 262 \\ 6.2.2 Geometric infinite divisibility / 263 \\ 6.2.3 Bernstein functions / 264 \\ 6.2.4 Self-decomposability / 265 \\ 6.2.5 Stable distributions / 265 \\ 6.2.6 Geometrically strictly stable distributions / 266 \\ 6.2.7 Mittag-Leffler distribution / 266 \\ 6.2.8 $\alpha$-Laplace distribution / 267 \\ 6.2.9 Semi-Pareto distribution / 268 \\ Exercises 6.2 / 268 \\ 6.3 Stationary Time Series Models / 269 \\ 6.3.1 Introduction / 269 \\ 6.3.2 Autoregressive models / 270 \\ 6.3.3 A general solution / 270 \\ 6.3.4 Extension to a $k$-th order autoregressive model / 272 \\ 6.3.5 Mittag-Leffler autoregressive structure / 273 \\ Exercises 6.3 / 274 \\ 6.4 A Structural Relationship and New Processes / 275 \\ 6.4.1 The TMLAR(1) process / 276 \\ 6.4.2 The NEAR(1) model / 277 \\ 6.4.3 New Mittag-Leffler autoregressive models / 278 \\ 6.4.4 The NSMLAR(1) process / 280 \\ Exercises 6.4 / 281 \\ 6.5 Tailed Processes / 281 \\ 6.5.1 The exponential tailed autoregressive process [ETAR(1)] / 282 \\ 6.5.2 The Mittag-Leffler tailed autoregressive process [MLTAR(1)] / 283 \\ Exercises 6.5 / 286 \\ 6.6 Marshall--Olkin Weibull Time Series Models / 286 \\ 6.6.1 Introduction / 286 \\ 6.6.2 Marshall--Olkin semi-Weibull distribution / 287 \\ 6.6.3 An AR(1) model with MOSW marginal distribution / 289 \\ 6.6.4 Marshall--Olkin generalized Weibull distribution / 291 \\ 6.6.5 An AR(1) model with MOGW marginal distribution / 292 \\ 6.6.6 Case study / 293 \\ Exercises 6.6 / 293 \\ References / 294 \\ 7 Applications to Density Estimation / 297 \\ 7.0 Density Estimation and Orthogonal Polynomials / 297 \\ 7.1 Introduction / 297 \\ 7.2 Approximants Based on Legendre Polynomials / 299 \\ 7.3 Approximants Based on Laguerre Polynomials / 301 \\ 7.4 A Unified Methodology / 304 \\ 7.5 Approximants Expressed in Terms of Orthogonal Polynomials / 305 \\ 7.5.1 Approximants expressed in terms of Laguerre polynomials / 306 \\ 7.5.2 Approximants expressed in terms of Legendre polynomials / 306 \\ 7.5.3 Approximants expressed in terms of Jacobi polynomials / 307 \\ 7.5.4 Approximants expressed in terms of Hermite polynomials / 307 \\ References / 308 \\ 8 Applications to Order Statistics / 311 \\ 8.0 Introduction / 311 \\ 8.1 Distribution Function / 311 \\ 8.1.1 Density of the $r$-th order statistic / 313 \\ 8.1.2 Joint distribution function of two order statistics / 314 \\ 8.1.3 Joint density of two order statistics / 314 \\ 8.1.4 Moments of order statistics / 315 \\ 8.1.5 Recurrence relations for moments / 317 \\ 8.1.6 Recurrence relations on the product moments / 318 \\ 8.1.7 Order statistics from symmetric distributions / 319 \\ 8.2 Discrete Order Statistics / 319 \\ 8.2.1 Probability function of discrete order statistics / 320 \\ 8.2.2 Joint probability function of two order statistics / 321 \\ 8.2.3 Bernoulli order statistics / 321 \\ 8.3 Independent Random Variables / 322 \\ 8.3.1 Distribution of a single order statistic / 322 \\ 8.3.2 Joint distribution of two order statistics / 324 \\ Test on Order Statistics / 325 \\ 8.4 On Concomitants of Order Statistics / 326 \\ 8.4.1 Application of concomitants of order statistics / 326 \\ 8.4.2 Application in estimation / 328 \\ 8.4.3 Concomitants of record values and estimation problems / 333 \\ References / 339 \\ 9 Applications to Astrophysics Problems / 341 \\ 9.0 Introduction / 341 \\ 9.1 Entropy: Boltzmann, Planck, and Einstein on W / 342 \\ 9.1.1 Entropic functional / 342 \\ 9.1.2 Entropy and probability / 342 \\ 9.1.3 Boltzmann--Gibbs / 343 \\ 9.2 Gravitationally Stabilized Fusion Reactor: The Sun / 345 \\ 9.2.1 Internal solar structure / 345 \\ 9.2.2 Solar fusion plasma / 349 \\ 9.2.3 Estimation of central temperature in the Sun / 349 \\ 9.3 Crucial Astrophysical Experiments: Data Analysis / 351 \\ 9.3.1 The experiments / 351 \\ 9.3.2 Analysis of the time series / 352 \\ 9.4 Fundamental Equations for Nonequilibrium Processes / 355 \\ 9.4.1 Chapman--Kolmogorov equation / 355 \\ 9.4.2 Master equation / 355 \\ 9.4.3 Fokker--Planck equation / 356 \\ 9.4.4 Langevin equation / 357 \\ 9.4.5 Reaction-diffusion equation / 357 \\ 9.5 Fractional Calculus / 360 \\ 9.6 Nonextensive Statistical Mechanics / 362 \\ 9.7 Standard and Fractional Reaction / 363 \\ 9.7.1 Boltzmann--Gibbs statistical mechanics / 363 \\ 9.7.2 Generalized Boltzmann--Gibbs statistical mechanics / 366 \\ 9.7.3 Fractional reaction / 369 \\ 9.7.4 Thermonuclear reaction coefficient / 371 \\ 9.8 Standard and Fractional Diffusion / 380 \\ 9.8.1 Fick's first law of diffusion / 380 \\ 9.8.2 Einstein's approach to diffusion / 381 \\ 9.8.3 Fractional diffusion / 381 \\ 9.8.4 Spatio-temporal pattern formation / 383 \\ References / 384 \\ 10 An Introduction to Wavelet Analysis / 389 \\ 10.0 Introduction / 389 \\ 10.1 Fourier Analysis to Wavelet Analysis / 390 \\ 10.2 Construction of Orthonormal Wavelets / 393 \\ 10.3 Classification of Wavelets and Multiresolution Analysis / 400 \\ 10.4 Spline Wavelets / 405 \\ 10.5 A Variant of Construction of Orthonormal Wavelets / 407 \\ Exercises 10.5 / 408 \\ References / 408 \\ 11 Jacobians of Matrix Transformations / 409 \\ 11.0 Introduction / 409 \\ 11.1 Jacobians of Linear Matrix Transformations / 410 \\ Exercises 11.1 / 415 \\ 11.2 Jacobians in Some Nonlinear Transformations / 417 \\ Exercises 11.2 / 421 \\ 11.3 Transformations Involving Orthonormal Matrices / 423 \\ Exercises 11.3 / 426 \\ References / 428 \\ 12 Special Functions of Matrix Argument / 429 \\ 12.0 Introduction / 429 \\ 12.1 Real Matrix-Variate Scalar Functions / 429 \\ 12.1.1 Real matrix-variate gamma / 430 \\ 12.1.2 Real matrix-variate gamma density / 430 \\ Exercises 12.1 / 434 \\ 12.2 The Laplace Transform in the Matrix Case / 435 \\ 12.2.1 A convolution property for Laplace transforms / 436 \\ Exercises 12.2 / 439 \\ 12.3 Hypergeometric Functions with Matrix Argument / 439 \\ 12.3.1 Hypergeometric function through Laplace transform / 440 \\ 12.3.2 Hypergeometric function through zonal polynomials / 441 \\ 12.3.3 Hypergeometric functions through $M$-transforms / 443 \\ 12.3.4 A convolution theorem for $M$-transforms / 445 \\ Exercises 12.3 / 446 \\ 12.4 A Pathway Model / 448 \\ 12.4.1 The pathway density / 448 \\ 12.4.2 A general density / 451 \\ 12.4.3 Arbitrary moments / 451 \\ 12.4.4 Quadratic forms / 452 \\ 12.4.5 Generalized quadratic form / 452 \\ 12.4.6 Applications to random volumes / 453 \\ Exercises 12.4 / 453 \\ References / 454 \\ Author Index / 457 \\ Subject Index / 461

entry:
@Book{Mathai:2008:SFA,
  author =       "A. M. Mathai and H. J. Haubold",
  title =        "Special Functions for Applied Scientists",
  publisher =    "Springer Science+Business Media",
  address =      "New York, NY, USA",
  pages =        "xxv + 464",
  year =         "2008",
  DOI =          "https://doi.org/10.1007/978-0-387-75894-7",
  ISBN =         "0-387-75893-3",
  ISBN-13 =      "978-0-387-75893-0",
  LCCN =         "QA351 .M37X 2008",
  bibdate =      "Sat Oct 30 17:02:02 MDT 2010",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 prodorbis.library.yale.edu:7090/voyager",
  acknowledgement = ack-nhfb,
  shorttableofcontents = "Basic ideas of special functions and
                 statistical distributions \\
                 Mittag-Leffler functions and fractional calculus \\
                 An introduction to q-series \\
                 Ramanujan's theories of theta and elliptic functions
                 \\
                 Lie group and special functions \\
                 Applications to stochastic process and time series \\
                 Applications to density estimation \\
                 Applications to order statistics \\
                 Applications to astrophysics problems \\
                 An introduction to wavelet analysis \\
                 Jacobians of matrix formations \\
                 Special functions of matrix argument",
  subject =      "Functions, Special; Fractional calculus; Wavelets
                 (Mathematics)",
  tableofcontents = "1 Basic Ideas of Special Functions and Statistical
                 Distributions / 1 \\
                 1.0 Introduction / 1 \\
                 1.1 Gamma Function / 3 \\
                 1.1.1 Some basic properties of gamma functions / 4 \\
                 1.1.2 Wedge product and Jacobians of transformations /
                 6 \\
                 1.1.3 Multiplication formula for gamma functions / 8
                 \\
                 1.1.4 Asymptotic formula for a gamma function / 9 \\
                 1.1.5 Bernoulli polynomials / 9 \\
                 1.1.6 Some basic properties of generalized Bernoulli
                 polynomials / 9 \\
                 1.1.7 The first three generalized Bernoulli polynomials
                 / 10 \\
                 Exercises 1.1 / 11 \\
                 1.2 The Psi and Zeta Functions / 12 \\
                 1.2.1 Generalized zeta function / 12 \\
                 Exercises 1.2 / 13 \\
                 1.3 Integral Transforms / 14 \\
                 1.3.1 Mellin transform / 14 \\
                 1.3.2 Laplace transform / 17 \\
                 Exercises 1.3 / 18 \\
                 1.4 Some Statistical Preliminaries / 19 \\
                 Exercises 1.4 / 26 \\
                 1.5 Some Properties of Random Variables / 28 \\
                 1.5.1 Multivariate analogues / 30 \\
                 1.5.2 Marginal and conditional densities / 31 \\
                 Exercises 1.5 / 33 \\
                 1.6 Beta and Related Functions / 34 \\
                 1.6.1 Dirichlet integrals and Dirichlet densities / 37
                 \\
                 Exercises 1.6 / 42 \\
                 1.7 Hypergeometric Series / 42 \\
                 1.7.1 Evaluation of some contour integrals / 45 \\
                 1.7.2 Residues when several gammas are involved / 46
                 \\
                 Exercises 1.7 / 48 \\
                 1.8 Meijer's $G$-function / 49 \\
                 Exercises 1.8 / 54 \\
                 1.9 The $H$-function / 54 \\
                 Exercises 1.9 / 57 \\
                 1.10 Lauricella Functions and Appell's Functions / 58
                 \\
                 1.10.1 Some properties of $f_A$ / 59 \\
                 1.10.2 Some properties of $f_B$ / 60 \\
                 1.10.3 Some properties of $f_C$ / 60 \\
                 1.10.4 Some properties of $f_D$ / 61 \\
                 Exercises 1.10 / 64 \\
                 1.11 Special Functions as Solutions of Differential
                 Equations and Applications / 64 \\
                 1.11.0 Introduction / 64 \\
                 1.11.1 Sine, cosine and exponential functions / 64 \\
                 Exercises 1.11 / 66 \\
                 1.11.2 Linear second order differential equations / 66
                 \\
                 1.11.3 Hypergeometric function / 67 \\
                 Exercises 1.11 / 69 \\
                 1.11.4 Confluent hypergeometric function / 70 \\
                 Exercises 1.11 / 71 \\
                 1.11.5 Hermite polynomials / 71 \\
                 Exercises 1.11 / 72 \\
                 1.11.6 Bessel functions / 72 \\
                 Exercises 1.11 / 73 \\
                 1.11.7 Laguerre polynomial / 74 \\
                 1.11.8 Legendre polynomial / 74 \\
                 Exercises 1.11 / 74 \\
                 1.11.9 Generalized hypergeometric function / 75 \\
                 1.11.10 $G$-function / 76 \\
                 Exercises 1.11 / 77 \\
                 References / 77 \\
                 2 Mittag-Leffler Functions and Fractional Calculus / 79
                 \\
                 2.0 Introduction / 79 \\
                 2.1 Mittag-Leffler Function / 79 \\
                 Revision Exercises 2.1 / 81 \\
                 2.2 Basic Properties of Mittag-Leffler Function / 82
                 \\
                 2.2.1 Mittag-Leffler functions of rational order / 84
                 \\
                 2.2.2 Euler transform of Mittag-Leffler function / 84
                 \\
                 2.2.3 Laplace transform of Mittag-Leffler function / 85
                 \\
                 2.2.4 Application of Laplace transform / 87 \\
                 2.2.5 Mittag-Leffler functions and the $H$-function /
                 88 \\
                 Exercises 2.2 / 90 \\
                 2.3 Generalized Mittag-Leffler Function / 91 \\
                 2.3.1 Special cases of $E_{\beta, \gamma}^\delta (z)$ /
                 92 \\
                 2.3.2 Mellin--Barnes integral representation / 92 \\
                 2.3.3 Relations with the H-function and Wright function
                 / 92 \\
                 2.3.4 Cases of reducibility / 93 \\
                 2.3.5 Differentiation of generalized Mittag-Leffler
                 function / 94 \\
                 2.3.6 Integral property of generalized Mittag-Leffler
                 function / 94 \\
                 2.3.7 Integral transform of $E_{\beta, \gamma}^\delta
                 (z)$ / 95 \\
                 Exercises 2.3 / 96 \\
                 2.4 Fractional Integrals / 97 \\
                 2.4.1 Riemann--Liouville fractional integrals of
                 arbitrary order / 98 \\
                 2.4.2 Riemann--Liouville fractional integrals of order
                 $\alpha$ / 99 \\
                 2.4.3 Basic properties of fractional integrals / 100
                 \\
                 2.4.4 A useful integral / 101 \\
                 2.4.5 The Weyl integral / 103 \\
                 2.4.6 Basic properties of Weyl integral / 104 \\
                 Exercises 2.4 / 105 \\
                 2.4.7 Laplace transform of the fractional integral /
                 107 \\
                 2.4.8 Laplace transform of the fractional derivative /
                 107 \\
                 2.4.9 Laplace transform of Caputo derivative / 108 \\
                 Exercises 2.4 / 109 \\
                 2.5 Mellin Transform of the Fractional Integrals and
                 the Fractional Derivatives / 109 \\
                 2.5.1 Mellin transform / 109 \\
                 2.5.2 Mellin transform of the fractional integral / 110
                 \\
                 2.5.3 Mellin transform of the fractional derivative /
                 111 \\
                 Exercises 2.5 / 111 \\
                 2.6 Kober Operators / 111 \\
                 Exercises 2.6 / 113 \\
                 2.7 Generalized Kober Operators / 114 \\
                 Exercises 2.7 / 118 \\
                 2.8 Compositions of Riemann--Liouville Fractional
                 Calculus Operators and Generalized Mittag-Leffler
                 Functions / 121 \\
                 2.8.1 Composition relations between R-L operators and
                 $E_{\beta, \gamma}^\delta (z)$ / 121 \\
                 Exercises 2.8 / 126 \\
                 2.9 Fractional Differential Equations / 126 \\
                 2.9.1 Fractional relaxation / 127 \\
                 Exercises 2.9 / 130 \\
                 2.9.2 Fractional diffusion / 131 \\
                 Exercises 2.9 / 132 \\
                 References / 133 \\
                 3 An Introduction to $q$-Series / 135 \\
                 3.0 Introduction / 135 \\
                 3.1 Hypergeometric Series / 135 \\
                 Exercises 3.1 / 136 \\
                 3.2 Basic Hypergeometric Series ($q$-Series) / 138 \\
                 Exercises 3.2 / 140 \\
                 3.2.1 The $q$-binomial theorem / 142 \\
                 Exercises 3.2 / 146 \\
                 3.2.2 The $q$-binomial coefficients / 146 \\
                 Exercises 3.2 / 148 \\
                 3.3 $q$-Calculus / 149 \\
                 Exercises 3.3 / 151 \\
                 3.4 The $q$-Gamma and $q$-Beta Functions / 151 \\
                 3.5 Transformation and Summation Formulas for
                 $q$-Series / 152 \\
                 Exercises 3.5 / 154 \\
                 3.6 Jacobi's Triple Product and Rogers--Ramanujan
                 Identities / 154 \\
                 References / 157 \\
                 4 Ramanujan's Theories of Theta and Elliptic Functions
                 / 159 \\
                 4.0 Introduction / 159 \\
                 4.1 Ramanujan's Theory of Classical Theta Functions /
                 159 \\
                 4.1.1 Series definition and additive results / 159 \\
                 4.2 Ramanujan's $_1\Psi_1$ Summation Formula and
                 Multiplicative Results for Theta Functions / 163 \\
                 4.3 Modular Equations / 170 \\
                 4.4 Inversion Formulas and Evaluations / 172 \\
                 4.5 Modular Identities (Classical Theory) / 175 \\
                 4.6 Ramanujan's Theory of Cubic Theta Functions / 177
                 \\
                 4.6.1 The cubic theta functions / 177 \\
                 4.6.2 Inversion formulas and evaluations (cubic theory)
                 / 179 \\
                 4.6.3 Triplication and trimediation formulas / 182 \\
                 4.6.4 Further evaluations / 183 \\
                 4.6.5 Evaluations of Ramanujan--Eisenstein series ($L$,
                 $M$, $N$ or $P$, $Q$, $R$) / 185 \\
                 4.6.6 The cubic analogue of the Jacobian elliptic
                 functions / 186 \\
                 Test on Ramanujan's work / 187 \\
                 4.7 The One-variable Cubic Theta Functions / 188 \\
                 4.7.1 Cubic theta functions and some properties / 189
                 \\
                 4.7.2 Product representations for $b(q)$ and $c(q)$ /
                 190 \\
                 4.7.3 The cubic analogue of Jacobi's quartic modular
                 equations / 191 \\
                 4.8 The Two-variable Cubic Theta Functions / 193 \\
                 4.8.1 Series definitions and some properties / 193 \\
                 4.8.2 Product representations for $b(q,z)$ and $c(q,z)$
                 / 195 \\
                 4.8.3 A two-variable cubic counterpart of Jacobi's
                 quartic modular equation / 200 \\
                 4.9 The Three-variable Cubic Theta Functions / 200 \\
                 4.9.1 Unification of one and two-variable cubic theta
                 functions / 200 \\
                 Exercises 4.9 / 200 \\
                 4.9.2 Generalization of Hirschhorn--Garvan--Borwein
                 identity / 202 \\
                 4.9.3 Laurent's expansions for two-parameter cubic
                 theta functions / 205 \\
                 References / 209 \\
                 5 Lie Group and Special Functions / 211 \\
                 5.1 General Introduction to Group Theory / 211 \\
                 5.1.1 Isomorphisms / 213 \\
                 5.1.2 Symmetry groups / 213 \\
                 5.1.3 Isometries of the Euclidean plane / 214 \\
                 5.1.4 Finite groups of motion / 215 \\
                 5.1.5 Discrete groups of motions / 216 \\
                 5.2 Lie Group and Special Functions / 217 \\
                 5.2.1 Subspace of a vector space / 218 \\
                 5.3 Lie Algebra / 219 \\
                 5.4 Representations of Lie Algebra / 222 \\
                 Exercises 5.4 / 225 \\
                 5.5 Special Functions / 226 \\
                 5.5.1 Gauss hypergeometric function / 226 \\
                 5.5.2 Differential equation satisfied by $_2F_1$ / 228
                 \\
                 5.5.3 Integral representation of
                 $_F_q(\alpha_1,\alpha_2, \ldots{}, \alpha_p \beta_1,
                 \beta_2, \ldots{}, \beta_p; z)$ / 230 \\
                 5.6 Laguerre Polynomial $L^{(\alpha)}_n(x)$ / 231 \\
                 5.6.1 Laguerre polynomial and Lie algebra / 232 \\
                 Exercises 5.6 / 232 \\
                 5.7 Helmholtz Equation / 234 \\
                 5.7.1 Helmholtz equation in three variables / 238 \\
                 5.8 Lie Group / 241 \\
                 5.8.1 Basis of the Lie algebra of the Lie group SL(2) /
                 242 \\
                 Exercises 5.8 / 243 \\
                 Test on Lie Theory and Special Functions / 244 \\
                 6 Applications to Stochastic Process and Time Series /
                 247 \\
                 6.0 Introduction / 247 \\
                 6.1 Stochastic Processes / 247 \\
                 6.1.1 Classical types of stochastic processes / 252 \\
                 6.1.2 Processes with stationary independent increments
                 / 252 \\
                 6.1.3 Stationary processes / 253 \\
                 6.1.4 Gaussian processes and stationarity / 253 \\
                 6.1.5 Brownian processes / 257 \\
                 6.1.6 Markov chains / 258 \\
                 Exercises 6.1 / 261 \\
                 6.2 Modern Concepts in Distribution Theory / 262 \\
                 6.2.1 Introduction / 262 \\
                 6.2.2 Geometric infinite divisibility / 263 \\
                 6.2.3 Bernstein functions / 264 \\
                 6.2.4 Self-decomposability / 265 \\
                 6.2.5 Stable distributions / 265 \\
                 6.2.6 Geometrically strictly stable distributions / 266
                 \\
                 6.2.7 Mittag-Leffler distribution / 266 \\
                 6.2.8 $\alpha$-Laplace distribution / 267 \\
                 6.2.9 Semi-Pareto distribution / 268 \\
                 Exercises 6.2 / 268 \\
                 6.3 Stationary Time Series Models / 269 \\
                 6.3.1 Introduction / 269 \\
                 6.3.2 Autoregressive models / 270 \\
                 6.3.3 A general solution / 270 \\
                 6.3.4 Extension to a $k$-th order autoregressive model
                 / 272 \\
                 6.3.5 Mittag-Leffler autoregressive structure / 273 \\
                 Exercises 6.3 / 274 \\
                 6.4 A Structural Relationship and New Processes / 275
                 \\
                 6.4.1 The TMLAR(1) process / 276 \\
                 6.4.2 The NEAR(1) model / 277 \\
                 6.4.3 New Mittag-Leffler autoregressive models / 278
                 \\
                 6.4.4 The NSMLAR(1) process / 280 \\
                 Exercises 6.4 / 281 \\
                 6.5 Tailed Processes / 281 \\
                 6.5.1 The exponential tailed autoregressive process
                 [ETAR(1)] / 282 \\
                 6.5.2 The Mittag-Leffler tailed autoregressive process
                 [MLTAR(1)] / 283 \\
                 Exercises 6.5 / 286 \\
                 6.6 Marshall--Olkin Weibull Time Series Models / 286
                 \\
                 6.6.1 Introduction / 286 \\
                 6.6.2 Marshall--Olkin semi-Weibull distribution / 287
                 \\
                 6.6.3 An AR(1) model with MOSW marginal distribution /
                 289 \\
                 6.6.4 Marshall--Olkin generalized Weibull distribution
                 / 291 \\
                 6.6.5 An AR(1) model with MOGW marginal distribution /
                 292 \\
                 6.6.6 Case study / 293 \\
                 Exercises 6.6 / 293 \\
                 References / 294 \\
                 7 Applications to Density Estimation / 297 \\
                 7.0 Density Estimation and Orthogonal Polynomials / 297
                 \\
                 7.1 Introduction / 297 \\
                 7.2 Approximants Based on Legendre Polynomials / 299
                 \\
                 7.3 Approximants Based on Laguerre Polynomials / 301
                 \\
                 7.4 A Unified Methodology / 304 \\
                 7.5 Approximants Expressed in Terms of Orthogonal
                 Polynomials / 305 \\
                 7.5.1 Approximants expressed in terms of Laguerre
                 polynomials / 306 \\
                 7.5.2 Approximants expressed in terms of Legendre
                 polynomials / 306 \\
                 7.5.3 Approximants expressed in terms of Jacobi
                 polynomials / 307 \\
                 7.5.4 Approximants expressed in terms of Hermite
                 polynomials / 307 \\
                 References / 308 \\
                 8 Applications to Order Statistics / 311 \\
                 8.0 Introduction / 311 \\
                 8.1 Distribution Function / 311 \\
                 8.1.1 Density of the $r$-th order statistic / 313 \\
                 8.1.2 Joint distribution function of two order
                 statistics / 314 \\
                 8.1.3 Joint density of two order statistics / 314 \\
                 8.1.4 Moments of order statistics / 315 \\
                 8.1.5 Recurrence relations for moments / 317 \\
                 8.1.6 Recurrence relations on the product moments / 318
                 \\
                 8.1.7 Order statistics from symmetric distributions /
                 319 \\
                 8.2 Discrete Order Statistics / 319 \\
                 8.2.1 Probability function of discrete order statistics
                 / 320 \\
                 8.2.2 Joint probability function of two order
                 statistics / 321 \\
                 8.2.3 Bernoulli order statistics / 321 \\
                 8.3 Independent Random Variables / 322 \\
                 8.3.1 Distribution of a single order statistic / 322
                 \\
                 8.3.2 Joint distribution of two order statistics / 324
                 \\
                 Test on Order Statistics / 325 \\
                 8.4 On Concomitants of Order Statistics / 326 \\
                 8.4.1 Application of concomitants of order statistics /
                 326 \\
                 8.4.2 Application in estimation / 328 \\
                 8.4.3 Concomitants of record values and estimation
                 problems / 333 \\
                 References / 339 \\
                 9 Applications to Astrophysics Problems / 341 \\
                 9.0 Introduction / 341 \\
                 9.1 Entropy: Boltzmann, Planck, and Einstein on W / 342
                 \\
                 9.1.1 Entropic functional / 342 \\
                 9.1.2 Entropy and probability / 342 \\
                 9.1.3 Boltzmann--Gibbs / 343 \\
                 9.2 Gravitationally Stabilized Fusion Reactor: The Sun
                 / 345 \\
                 9.2.1 Internal solar structure / 345 \\
                 9.2.2 Solar fusion plasma / 349 \\
                 9.2.3 Estimation of central temperature in the Sun /
                 349 \\
                 9.3 Crucial Astrophysical Experiments: Data Analysis /
                 351 \\
                 9.3.1 The experiments / 351 \\
                 9.3.2 Analysis of the time series / 352 \\
                 9.4 Fundamental Equations for Nonequilibrium Processes
                 / 355 \\
                 9.4.1 Chapman--Kolmogorov equation / 355 \\
                 9.4.2 Master equation / 355 \\
                 9.4.3 Fokker--Planck equation / 356 \\
                 9.4.4 Langevin equation / 357 \\
                 9.4.5 Reaction-diffusion equation / 357 \\
                 9.5 Fractional Calculus / 360 \\
                 9.6 Nonextensive Statistical Mechanics / 362 \\
                 9.7 Standard and Fractional Reaction / 363 \\
                 9.7.1 Boltzmann--Gibbs statistical mechanics / 363 \\
                 9.7.2 Generalized Boltzmann--Gibbs statistical
                 mechanics / 366 \\
                 9.7.3 Fractional reaction / 369 \\
                 9.7.4 Thermonuclear reaction coefficient / 371 \\
                 9.8 Standard and Fractional Diffusion / 380 \\
                 9.8.1 Fick's first law of diffusion / 380 \\
                 9.8.2 Einstein's approach to diffusion / 381 \\
                 9.8.3 Fractional diffusion / 381 \\
                 9.8.4 Spatio-temporal pattern formation / 383 \\
                 References / 384 \\
                 10 An Introduction to Wavelet Analysis / 389 \\
                 10.0 Introduction / 389 \\
                 10.1 Fourier Analysis to Wavelet Analysis / 390 \\
                 10.2 Construction of Orthonormal Wavelets / 393 \\
                 10.3 Classification of Wavelets and Multiresolution
                 Analysis / 400 \\
                 10.4 Spline Wavelets / 405 \\
                 10.5 A Variant of Construction of Orthonormal Wavelets
                 / 407 \\
                 Exercises 10.5 / 408 \\
                 References / 408 \\
                 11 Jacobians of Matrix Transformations / 409 \\
                 11.0 Introduction / 409 \\
                 11.1 Jacobians of Linear Matrix Transformations / 410
                 \\
                 Exercises 11.1 / 415 \\
                 11.2 Jacobians in Some Nonlinear Transformations / 417
                 \\
                 Exercises 11.2 / 421 \\
                 11.3 Transformations Involving Orthonormal Matrices /
                 423 \\
                 Exercises 11.3 / 426 \\
                 References / 428 \\
                 12 Special Functions of Matrix Argument / 429 \\
                 12.0 Introduction / 429 \\
                 12.1 Real Matrix-Variate Scalar Functions / 429 \\
                 12.1.1 Real matrix-variate gamma / 430 \\
                 12.1.2 Real matrix-variate gamma density / 430 \\
                 Exercises 12.1 / 434 \\
                 12.2 The Laplace Transform in the Matrix Case / 435 \\
                 12.2.1 A convolution property for Laplace transforms /
                 436 \\
                 Exercises 12.2 / 439 \\
                 12.3 Hypergeometric Functions with Matrix Argument /
                 439 \\
                 12.3.1 Hypergeometric function through Laplace
                 transform / 440 \\
                 12.3.2 Hypergeometric function through zonal
                 polynomials / 441 \\
                 12.3.3 Hypergeometric functions through $M$-transforms
                 / 443 \\
                 12.3.4 A convolution theorem for $M$-transforms / 445
                 \\
                 Exercises 12.3 / 446 \\
                 12.4 A Pathway Model / 448 \\
                 12.4.1 The pathway density / 448 \\
                 12.4.2 A general density / 451 \\
                 12.4.3 Arbitrary moments / 451 \\
                 12.4.4 Quadratic forms / 452 \\
                 12.4.5 Generalized quadratic form / 452 \\
                 12.4.6 Applications to random volumes / 453 \\
                 Exercises 12.4 / 453 \\
                 References / 454 \\
                 Author Index / 457 \\
                 Subject Index / 461",
}


bibtype: article
filename: elefunt.bib
label: Nam:2008:PAE
author: Byeong-Gyu Nam and Hyejung Kim and Hoi-Jun Yoo
journal: j-IEEE-TRANS-COMPUT
volume: 57
number: 4
pages: 490--504
month: apr
monthnumber: 4
year: 2008
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2008.12
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Mon Jul 4 12:17:40 MDT 2011
bibtimestamp: 2011.07.04 12:17:40 MDT
URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4432232
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: Byeong-Gyu Nam and Hyejung Kim and Hoi-Jun Yoo
title:
Power and Area-Efficient Unified Computation of Vector and Elementary Functions for Handheld $3$D Graphics Systems

abstract:
A unified computation method of vector and elementary functions is proposed for handheld 3D graphics systems. It unifies vector operations like vector multiply, multiply-and-add, divide, divide-by-square-root, and dot product and elementary functions like trigonometric, inverse trigonometric, hyperbolic, inverse hyperbolic, power ($ x^y $ with two variables), and logarithm to an arbitrary base into a single four-way arithmetic platform. A number system called the fixed-point hybrid number system (FXP-HNS), which combines the fixed-point number system (FXP) and the logarithmic number system (LNS), is proposed for the power and area-efficient unification. Power and area-efficient logarithmic and antilogarithmic conversion schemes are also proposed for the data conversions between fixed-point and logarithmic numbers in the FXP-HNS and achieve 0.41 percent and 0.08 percent maximum conversion errors, respectively. The unified arithmetic unit based on the proposed schemes is presented with less than 6.3 percent operation error. Its fully pipelined architecture achieves single-cycle throughput with maximum four-cycle latency for all of the supported operations. Comparison results show that the proposed arithmetic unit achieves 30 percent power and 10.9 percent area reductions and runs two times faster than the previous approach.

rawabstract:
A unified computation method of vector and elementary functions is proposed for handheld 3D graphics systems. It unifies vector operations like vector multiply, multiply-and-add, divide, divide-by-square-root, and dot product and elementary functions like trigonometric, inverse trigonometric, hyperbolic, inverse hyperbolic, power ($ x^y $ with two variables), and logarithm to an arbitrary base into a single four-way arithmetic platform. A number system called the fixed-point hybrid number system (FXP-HNS), which combines the fixed-point number system (FXP) and the logarithmic number system (LNS), is proposed for the power and area-efficient unification. Power and area-efficient logarithmic and antilogarithmic conversion schemes are also proposed for the data conversions between fixed-point and logarithmic numbers in the FXP-HNS and achieve 0.41 percent and 0.08 percent maximum conversion errors, respectively. The unified arithmetic unit based on the proposed schemes is presented with less than 6.3 percent operation error. Its fully pipelined architecture achieves single-cycle throughput with maximum four-cycle latency for all of the supported operations. Comparison results show that the proposed arithmetic unit achieves 30 percent power and 10.9 percent area reductions and runs two times faster than the previous approach.

rawtitle:
Power and Area-Efficient Unified Computation of Vector and Elementary Functions for Handheld {$3$D} Graphics Systems

entry:
@Article{Nam:2008:PAE,
  author =       "Byeong-Gyu Nam and Hyejung Kim and Hoi-Jun Yoo",
  title =        "Power and Area-Efficient Unified Computation of Vector
                 and Elementary Functions for Handheld {$3$D} Graphics
                 Systems",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "57",
  number =       "4",
  pages =        "490--504",
  month =        apr,
  year =         "2008",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2008.12",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Mon Jul 4 12:17:40 MDT 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib",
  URL =          "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4432232",
  abstract =     "A unified computation method of vector and elementary
                 functions is proposed for handheld 3D graphics systems.
                 It unifies vector operations like vector multiply,
                 multiply-and-add, divide, divide-by-square-root, and
                 dot product and elementary functions like
                 trigonometric, inverse trigonometric, hyperbolic,
                 inverse hyperbolic, power ($ x^y $ with two variables),
                 and logarithm to an arbitrary base into a single
                 four-way arithmetic platform. A number system called
                 the fixed-point hybrid number system (FXP-HNS), which
                 combines the fixed-point number system (FXP) and the
                 logarithmic number system (LNS), is proposed for the
                 power and area-efficient unification. Power and
                 area-efficient logarithmic and antilogarithmic
                 conversion schemes are also proposed for the data
                 conversions between fixed-point and logarithmic numbers
                 in the FXP-HNS and achieve 0.41 percent and 0.08
                 percent maximum conversion errors, respectively. The
                 unified arithmetic unit based on the proposed schemes
                 is presented with less than 6.3 percent operation
                 error. Its fully pipelined architecture achieves
                 single-cycle throughput with maximum four-cycle latency
                 for all of the supported operations. Comparison results
                 show that the proposed arithmetic unit achieves 30
                 percent power and 10.9 percent area reductions and runs
                 two times faster than the previous approach.",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
}


bibtype: article
filename: elefunt.bib
label: Paszkowski:2008:CAO
author: Stefan Paszkowski
title: Convergence acceleration of orthogonal series
journal: j-NUMER-ALGORITHMS
volume: 47
number: 1
pages: 35--62
month: jan
monthnumber: 1
year: 2008
CODEN: NUALEG
DOI: https://doi.org/10.1007/s11075-007-9146-7
ISSN: 1017-1398 (print), 1572-9265 (electronic) OR 10171398 (print), 15729265 (electronic)
ISSNL: 1017-1398 OR 10171398
MRclass: subject classification (2000); 33C45; 42A32; 42C10; 42C20; 65B10
bibdate: Tue Jul 8 19:14:30 MDT 2008
bibtimestamp: 2008.07.08 19:14:30 MDT
URL: http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=47&issue=1&spage=35
fjournal: Numerical Algorithms
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11075
rawauthor: Stefan Paszkowski
rawtitle: Convergence acceleration of orthogonal series
bibsource:
http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1017-1398&volume=47&issue=1; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

keywords:
Convergence acceleration; convergence acceleration; Orthogonal polynomials; Orthogonal series; Trigonometric series

entry:
@Article{Paszkowski:2008:CAO,
  author =       "Stefan Paszkowski",
  title =        "Convergence acceleration of orthogonal series",
  journal =      j-NUMER-ALGORITHMS,
  volume =       "47",
  number =       "1",
  pages =        "35--62",
  month =        jan,
  year =         "2008",
  CODEN =        "NUALEG",
  DOI =          "https://doi.org/10.1007/s11075-007-9146-7",
  ISSN =         "1017-1398 (print), 1572-9265 (electronic)",
  ISSN-L =       "1017-1398",
  MRclass =      "subject classification (2000); 33C45; 42A32; 42C10;
                 42C20; 65B10",
  bibdate =      "Tue Jul 8 19:14:30 MDT 2008",
  bibsource =    "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1017-1398&volume=47&issue=1;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=47&issue=1&spage=35",
  acknowledgement = ack-nhfb,
  fjournal =     "Numerical Algorithms",
  journal-URL =  "http://link.springer.com/journal/11075",
  keywords =     "Convergence acceleration; convergence acceleration;
                 Orthogonal polynomials; Orthogonal series;
                 Trigonometric series",
}


bibtype: article
filename: elefunt.bib
label: Pinchon:2008:NEL
author: Didier Pinchon and Philip E. Hoggan and Frank E. Harris
title: A new expansion of the leaky aquifer function
journal: j-IJQC
volume: 108
number: 15
pages: 3042--3046
year: 2008
CODEN: IJQCB2
DOI: https://doi.org/10.1002/qua.21448; https://doi.org/10.1002/qua.21835
ISSN: 0020-7608 (print), 1097-461X (electronic) OR 00207608 (print), 1097461X (electronic)
ISSNL: 0020-7608 OR 00207608
MRclass: 86A05 80A20 33C10 82B80
bibdate: Sat Oct 1 14:02:23 MDT 2011
bibtimestamp: 2011.10.01 14:02:23 MDT
fjournal: International Journal of Quantum Chemistry
ZMnumber: Zbl 1189.86005
acknowledgement: ack-nhfb
ajournal: Int. J. Quantum Chem.
journalURL: http://www.interscience.wiley.com/jpages/0020-7608/
onlinedate: 4 Aug 2008
rawauthor: Didier Pinchon and Philip E. Hoggan and Frank E. Harris
rawtitle: A new expansion of the leaky aquifer function
bibsource:
https://www.math.utah.edu/pub/bibnet/authors/h/harris-frank-e.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/ijqc2000.bib

entry:
@Article{Pinchon:2008:NEL,
  author =       "Didier Pinchon and Philip E. Hoggan and Frank E.
                 Harris",
  title =        "A new expansion of the leaky aquifer function",
  journal =      j-IJQC,
  volume =       "108",
  number =       "15",
  pages =        "3042--3046",
  month =        "????",
  year =         "2008",
  CODEN =        "IJQCB2",
  DOI =          "https://doi.org/10.1002/qua.21448;
                 https://doi.org/10.1002/qua.21835",
  ISSN =         "0020-7608 (print), 1097-461X (electronic)",
  ISSN-L =       "0020-7608",
  MRclass =      "86A05 80A20 33C10 82B80",
  bibdate =      "Sat Oct 1 14:02:23 MDT 2011",
  bibsource =    "https://www.math.utah.edu/pub/bibnet/authors/h/harris-frank-e.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/ijqc2000.bib",
  ZMnumber =     "Zbl 1189.86005",
  acknowledgement = ack-nhfb,
  ajournal =     "Int. J. Quantum Chem.",
  fjournal =     "International Journal of Quantum Chemistry",
  journal-URL =  "http://www.interscience.wiley.com/jpages/0020-7608/",
  onlinedate =   "4 Aug 2008",
}


bibtype: article
filename: elefunt.bib
label: Pineiro:2008:RDD
author: J.-A. Pineiro and J. D. Bruguera and F. Lamberti and P. Montuschi
title: A Radix-2 Digit-by-Digit Architecture for Cube Root
journal: j-IEEE-TRANS-COMPUT
volume: 57
number: 4
pages: 562--566
month: apr
monthnumber: 4
year: 2008
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2007.70848
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Mon Jul 4 12:17:41 MDT 2011
bibtimestamp: 2011.07.04 12:17:41 MDT
URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4407683
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: J.-A. Pineiro and J. D. Bruguera and F. Lamberti and P. Montuschi
rawtitle: A Radix-2 Digit-by-Digit Architecture for Cube Root
abstract:
A radix-2 digit-recurrence algorithm and architecture for the computation of the cube root are presented in this paper. The original recurrence based on the concept of completing the cube is modified to allow an efficient implementation of the algorithm and the cycle time and area cost of the resulting architecture are estimated as 7.5 times the delay of a full adder and around 9,000 nand2 cells, respectively, for double-precision computations.

rawabstract:
A radix-2 digit-recurrence algorithm and architecture for the computation of the cube root are presented in this paper. The original recurrence based on the concept of completing the cube is modified to allow an efficient implementation of the algorithm and the cycle time and area cost of the resulting architecture are estimated as 7.5 times the delay of a full adder and around 9,000 nand2 cells, respectively, for double-precision computations.

entry:
@Article{Pineiro:2008:RDD,
  author =       "J.-A. Pineiro and J. D. Bruguera and F. Lamberti and
                 P. Montuschi",
  title =        "A Radix-2 Digit-by-Digit Architecture for Cube Root",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "57",
  number =       "4",
  pages =        "562--566",
  month =        apr,
  year =         "2008",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2007.70848",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Mon Jul 4 12:17:41 MDT 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib",
  URL =          "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4407683",
  abstract =     "A radix-2 digit-recurrence algorithm and architecture
                 for the computation of the cube root are presented in
                 this paper. The original recurrence based on the
                 concept of completing the cube is modified to allow an
                 efficient implementation of the algorithm and the cycle
                 time and area cost of the resulting architecture are
                 estimated as 7.5 times the delay of a full adder and
                 around 9,000 nand2 cells, respectively, for
                 double-precision computations.",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
}


bibtype: inproceedings
filename: elefunt.bib
label: Piso:2008:NRA
author: D. Piso and J. D. Bruguera
editor: Luca Fanucci
title: A New Rounding Algorithm for Variable Latency Division and Square Root Implementations
publisher: pub-IEEE
address: pub-IEEE:adr
pages: 760--767
year: 2008
DOI: https://doi.org/10.1109/DSD.2008.28.
ISBN: 0-7695-3277-2 OR 0769532772
ISBN13: 978-0-7695-3277-6 OR 9780769532776
bibdate: Sun Dec 10 13:55:38 2023
bibtimestamp: 2023.12.10 13:55:38 ???
acknowledgement: ack-nhfb
rawauthor: D. Piso and J. D. Bruguera
raweditor: Luca Fanucci
rawtitle: A New Rounding Algorithm for Variable Latency Division and Square Root Implementations
booktitle:
Proceedings: 11th Euromicro Symposium on Digital Systems Design: Architectures, Methods and Tools (DSD 2008), Parma, Italy, September 3--5, 2008

abstract:
The aim of this work is to present a method for rounding quadratically converging algorithms that improves their performance. This method is able to reduce significantly the number of cases where the remainder calculation is necessary. It is based on previous methods and incorporates additional bits of the result approximation to be checked. This work includes the result of exhaustive simulations that permit us to measure exactly how many calculations are avoided. Using these simulations, it is concluded that the presented method is able to reduce by half the number of remainder calculations. Using adequate result approximations the remainder calculation is necessary in only 5% of the total cases

rawabstract:
The aim of this work is to present a method for rounding quadratically converging algorithms that improves their performance. This method is able to reduce significantly the number of cases where the remainder calculation is necessary. It is based on previous methods and incorporates additional bits of the result approximation to be checked. This work includes the result of exhaustive simulations that permit us to measure exactly how many calculations are avoided. Using these simulations, it is concluded that the presented method is able to reduce by half the number of remainder calculations. Using adequate result approximations the remainder calculation is necessary in only 5\% of the total cases

rawbooktitle:
Proceedings: {11th Euromicro Symposium on Digital Systems Design: Architectures, Methods and Tools (DSD 2008), Parma, Italy, September 3--5, 2008}

entry:
@InProceedings{Piso:2008:NRA,
  author =       "D. Piso and J. D. Bruguera",
  editor =       "Luca Fanucci",
  booktitle =    "Proceedings: {11th Euromicro Symposium on Digital
                 Systems Design: Architectures, Methods and Tools (DSD
                 2008), Parma, Italy, September 3--5, 2008}",
  title =        "A New Rounding Algorithm for Variable Latency Division
                 and Square Root Implementations",
  publisher =    pub-IEEE,
  address =      pub-IEEE:adr,
  pages =        "760--767",
  year =         "2008",
  DOI =          "https://doi.org/10.1109/DSD.2008.28.",
  ISBN =         "0-7695-3277-2",
  ISBN-13 =      "978-0-7695-3277-6",
  bibdate =      "Sun Dec 10 13:55:38 2023",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  abstract =     "The aim of this work is to present a method for
                 rounding quadratically converging algorithms that
                 improves their performance. This method is able to
                 reduce significantly the number of cases where the
                 remainder calculation is necessary. It is based on
                 previous methods and incorporates additional bits of
                 the result approximation to be checked. This work
                 includes the result of exhaustive simulations that
                 permit us to measure exactly how many calculations are
                 avoided. Using these simulations, it is concluded that
                 the presented method is able to reduce by half the
                 number of remainder calculations. Using adequate result
                 approximations the remainder calculation is necessary
                 in only 5\% of the total cases",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: elefunt.bib
label: Rodriguez-Henriquez:2008:LCB
author: F. Rodriguez-Henriquez and G. Morales-Luna and J. Lopez
title: Low-Complexity Bit-Parallel Square Root Computation over $ mathrm GF(2^m) $ for All Trinomials
journal: j-IEEE-TRANS-COMPUT
volume: 57
number: 4
pages: 472--480
month: apr
monthnumber: 4
year: 2008
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2007.70822
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Mon Jul 4 12:17:40 MDT 2011
bibtimestamp: 2011.07.04 12:17:40 MDT
URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4358282
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: F. Rodriguez-Henriquez and G. Morales-Luna and J. Lopez
rawtitle: Low-Complexity Bit-Parallel Square Root Computation over {$ \mathrm {GF}(2^m) $} for All Trinomials
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/ieeetranscomput2000.bib

entry:
@Article{Rodriguez-Henriquez:2008:LCB,
  author =       "F. Rodriguez-Henriquez and G. Morales-Luna and J.
                 Lopez",
  title =        "Low-Complexity Bit-Parallel Square Root Computation
                 over {$ \mathrm {GF}(2^m) $} for All Trinomials",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "57",
  number =       "4",
  pages =        "472--480",
  month =        apr,
  year =         "2008",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2007.70822",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Mon Jul 4 12:17:40 MDT 2011",
  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/ieeetranscomput2000.bib",
  URL =          "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4358282",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
}


bibtype: article
filename: elefunt.bib
label: Sablonniere:2008:BSH
author: Paul Sablonniere
title: B-splines and Hermite--Pade approximants to the exponential function
journal: j-J-COMPUT-APPL-MATH
volume: 219
number: 2
pages: 509--517
day: 1
month: oct
monthnumber: 10
year: 2008
CODEN: JCAMDI
ISSN: 0377-0427 (print), 1879-1778 (electronic) OR 03770427 (print), 18791778 (electronic)
ISSNL: 0377-0427 OR 03770427
bibdate: Sat Feb 25 13:13:26 MST 2017
bibtimestamp: 2017.02.25 13:13:26 MST
URL: http://www.sciencedirect.com/science/article/pii/S037704270700252X
fjournal: Journal of Computational and Applied Mathematics
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03770427
rawauthor: Paul Sablonni{\`e}re
rawtitle: {B}-splines and {Hermite--Pad{\'e}} approximants to the exponential function
entry:
@Article{Sablonniere:2008:BSH,
  author =       "Paul Sablonni{\`e}re",
  title =        "{B}-splines and {Hermite--Pad{\'e}} approximants to
                 the exponential function",
  journal =      j-J-COMPUT-APPL-MATH,
  volume =       "219",
  number =       "2",
  pages =        "509--517",
  day =          "1",
  month =        oct,
  year =         "2008",
  CODEN =        "JCAMDI",
  ISSN =         "0377-0427 (print), 1879-1778 (electronic)",
  ISSN-L =       "0377-0427",
  bibdate =      "Sat Feb 25 13:13:26 MST 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2005.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S037704270700252X",
  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: Schreier:2008:OIR
author: Franz Schreier and Dieter Kohlert
journal: j-COMP-PHYS-COMM
volume: 179
number: 7
pages: 457--465
day: 1
month: oct
monthnumber: 10
year: 2008
CODEN: CPHCBZ
DOI: https://doi.org/10.1016/j.cpc.2008.04.012
ISSN: 0010-4655 (print), 1879-2944 (electronic) OR 00104655 (print), 18792944 (electronic)
ISSNL: 0010-4655 OR 00104655
bibdate: Mon Feb 13 23:42:36 MST 2012
bibtimestamp: 2012.02.13 23:42:36 MST
URL: http://www.sciencedirect.com/science/article/pii/S0010465508001495
fjournal: Computer Physics Communications
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00104655
rawauthor: Franz Schreier and Dieter Kohlert
title:
Optimized implementations of rational approximations --- a case study on the Voigt and complex error function

bibsource:
https://www.math.utah.edu/pub/tex/bib/compphyscomm2000.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

rawtitle:
Optimized implementations of rational approximations --- a case study on the {Voigt} and complex error function

entry:
@Article{Schreier:2008:OIR,
  author =       "Franz Schreier and Dieter Kohlert",
  title =        "Optimized implementations of rational approximations
                 --- a case study on the {Voigt} and complex error
                 function",
  journal =      j-COMP-PHYS-COMM,
  volume =       "179",
  number =       "7",
  pages =        "457--465",
  day =          "1",
  month =        oct,
  year =         "2008",
  CODEN =        "CPHCBZ",
  DOI =          "https://doi.org/10.1016/j.cpc.2008.04.012",
  ISSN =         "0010-4655 (print), 1879-2944 (electronic)",
  ISSN-L =       "0010-4655",
  bibdate =      "Mon Feb 13 23:42:36 MST 2012",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/compphyscomm2000.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0010465508001495",
  acknowledgement = ack-nhfb,
  fjournal =     "Computer Physics Communications",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00104655",
}


bibtype: article
filename: elefunt.bib
label: Segura:2008:IZC
author: Javier Segura
title: Interlacing of the zeros of contiguous hypergeometric functions
journal: j-NUMER-ALGORITHMS
volume: 49
number: 1--4
pages: 387--407
month: dec
monthnumber: 12
year: 2008
CODEN: NUALEG
ISSN: 1017-1398 (print), 1572-9265 (electronic) OR 10171398 (print), 15729265 (electronic)
ISSNL: 1017-1398 OR 10171398
bibdate: Mon May 17 15:07:19 MDT 2010
bibtimestamp: 2010.05.17 15:07:19 MDT
URL: http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=49&issue=1&spage=387
fjournal: Numerical Algorithms
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11075
rawauthor: Javier Segura
rawtitle: Interlacing of the zeros of contiguous hypergeometric functions
bibsource:
http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1017-1398&volume=49&issue=1; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib

entry:
@Article{Segura:2008:IZC,
  author =       "Javier Segura",
  title =        "Interlacing of the zeros of contiguous hypergeometric
                 functions",
  journal =      j-NUMER-ALGORITHMS,
  volume =       "49",
  number =       "1--4",
  pages =        "387--407",
  month =        dec,
  year =         "2008",
  CODEN =        "NUALEG",
  ISSN =         "1017-1398 (print), 1572-9265 (electronic)",
  ISSN-L =       "1017-1398",
  bibdate =      "Mon May 17 15:07:19 MDT 2010",
  bibsource =    "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1017-1398&volume=49&issue=1;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/numeralgorithms.bib",
  URL =          "http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=49&issue=1&spage=387",
  acknowledgement = ack-nhfb,
  fjournal =     "Numerical Algorithms",
  journal-URL =  "http://link.springer.com/journal/11075",
}


bibtype: misc
filename: elefunt.bib
label: Steele:2008:FPSb
author: Guy L. Steele Jr.
title: Floating point square root provider with embedded status information
pages: 387--407
day: 30
month: sep
monthnumber: 9
year: 2008
bibdate: Tue Dec 23 15:06:43 2008
bibtimestamp: 2008.12.23 15:06:43 ???
URL: http://www.patentstorm.us/patents/7430576/fulltext.html
acknowledgement: ack-nhfb
howpublished: US Patent 7430576
rawauthor: Guy L. {Steele Jr.}
rawtitle: Floating point square root provider with embedded status information
abstract:
A system for providing a floating point square root comprises an analyzer circuit configured to determine a first status of a first floating point operand based upon data within the first floating point operand. In addition, the system comprises a results circuit coupled to the analyzer circuit. The results circuit is configured to assert a resulting floating point operand containing the square root of the first floating point operand and a resulting status embedded within the resulting floating point operand.

rawabstract:
A system for providing a floating point square root comprises an analyzer circuit configured to determine a first status of a first floating point operand based upon data within the first floating point operand. In addition, the system comprises a results circuit coupled to the analyzer circuit. The results circuit is configured to assert a resulting floating point operand containing the square root of the first floating point operand and a resulting status embedded within the resulting floating point operand.

entry:
@Misc{Steele:2008:FPSb,
  author =       "Guy L. {Steele Jr.}",
  title =        "Floating point square root provider with embedded
                 status information",
  howpublished = "US Patent 7430576",
  day =          "30",
  month =        sep,
  year =         "2008",
  bibdate =      "Tue Dec 23 15:06:43 2008",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://www.patentstorm.us/patents/7430576/fulltext.html",
  abstract =     "A system for providing a floating point square root
                 comprises an analyzer circuit configured to determine a
                 first status of a first floating point operand based
                 upon data within the first floating point operand. In
                 addition, the system comprises a results circuit
                 coupled to the analyzer circuit. The results circuit is
                 configured to assert a resulting floating point operand
                 containing the square root of the first floating point
                 operand and a resulting status embedded within the
                 resulting floating point operand.",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: elefunt.bib
label: Vepstas:2008:EAA
author: Linas Vepstas
journal: j-NUMER-ALGORITHMS
volume: 47
number: 3
pages: 211--252
month: mar
monthnumber: 3
year: 2008
CODEN: NUALEG
DOI: https://doi.org/10.1007/s11075-007-9153-8
ISSN: 1017-1398 (print), 1572-9265 (electronic) OR 10171398 (print), 15729265 (electronic)
ISSNL: 1017-1398 OR 10171398
bibdate: Mon May 17 15:49:34 MDT 2010
bibtimestamp: 2010.05.17 15:49:34 MDT
URL: http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=47&issue=3&spage=211
fjournal: Numerical Algorithms
keywords: convergence acceleration
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/11075
rawauthor: Linas Vepstas
title:
An efficient algorithm for accelerating the convergence of oscillatory series, useful for computing the polylogarithm and Hurwitz zeta functions

bibsource:
http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1017-1398&volume=47&issue=3; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

abstract:
This paper sketches a technique for improving the rate of convergence of a general oscillatory sequence, and then applies this series acceleration algorithm to the polylogarithm and the Hurwitz zeta function. As such, it may be taken as an extension of the techniques given by Borwein's ``An efficient algorithm for computing the Riemann zeta function'' by Borwein for computing the Riemann zeta function, to more general series. The algorithm provides a rapid means of evaluating $ operatorname Li_s(z) $ for general values of complex $s$ and a kidney-shaped region of complex $z$ values given by $ |z^2 / (z - 1)| < 4 $. By using the duplication formula and the inversion formula, the range of convergence for the polylogarithm may be extended to the entire complex $z$-plane, and so the algorithms described here allow for the evaluation of the polylogarithm for all complex $s$ and $z$ values. Alternatively, the Hurwitz zeta can be very rapidly evaluated by means of an Euler Maclaurin series. The polylogarithm and the Hurwitz zeta are related, in that two evaluations of the one can be used to obtain a value of the other; thus, either algorithm can be used to evaluate either function. The Euler Maclaurin series is a clear performance winner for the Hurwitz zeta, while the Borwein algorithm is superior for evaluating the polylogarithm in the kidney-shaped region. Both algorithms are superior to the simple Taylor's series or direct summation. The primary, concrete result of this paper is an algorithm allows the exploration of the Hurwitz zeta in the critical strip, where fast algorithms are otherwise unavailable. A discussion of the monodromy group of the polylogarithm is included.

rawabstract:
This paper sketches a technique for improving the rate of convergence of a general oscillatory sequence, and then applies this series acceleration algorithm to the polylogarithm and the Hurwitz zeta function. As such, it may be taken as an extension of the techniques given by Borwein's ``An efficient algorithm for computing the Riemann zeta function'' by Borwein for computing the Riemann zeta function, to more general series. The algorithm provides a rapid means of evaluating $ \operatorname {Li}_s(z) $ for general values of complex $s$ and a kidney-shaped region of complex $z$ values given by $ |z^2 / (z - 1)| < 4 $. By using the duplication formula and the inversion formula, the range of convergence for the polylogarithm may be extended to the entire complex $z$-plane, and so the algorithms described here allow for the evaluation of the polylogarithm for all complex $s$ and $z$ values. Alternatively, the Hurwitz zeta can be very rapidly evaluated by means of an Euler Maclaurin series. The polylogarithm and the Hurwitz zeta are related, in that two evaluations of the one can be used to obtain a value of the other; thus, either algorithm can be used to evaluate either function. The Euler Maclaurin series is a clear performance winner for the Hurwitz zeta, while the Borwein algorithm is superior for evaluating the polylogarithm in the kidney-shaped region. Both algorithms are superior to the simple Taylor's series or direct summation. The primary, concrete result of this paper is an algorithm allows the exploration of the Hurwitz zeta in the critical strip, where fast algorithms are otherwise unavailable. A discussion of the monodromy group of the polylogarithm is included.

rawtitle:
An efficient algorithm for accelerating the convergence of oscillatory series, useful for computing the polylogarithm and {Hurwitz} zeta functions

entry:
@Article{Vepstas:2008:EAA,
  author =       "Linas Vepstas",
  title =        "An efficient algorithm for accelerating the
                 convergence of oscillatory series, useful for computing
                 the polylogarithm and {Hurwitz} zeta functions",
  journal =      j-NUMER-ALGORITHMS,
  volume =       "47",
  number =       "3",
  pages =        "211--252",
  month =        mar,
  year =         "2008",
  CODEN =        "NUALEG",
  DOI =          "https://doi.org/10.1007/s11075-007-9153-8",
  ISSN =         "1017-1398 (print), 1572-9265 (electronic)",
  ISSN-L =       "1017-1398",
  bibdate =      "Mon May 17 15:49:34 MDT 2010",
  bibsource =    "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=1017-1398&volume=47&issue=3;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.springerlink.com/openurl.asp?genre=article&issn=1017-1398&volume=47&issue=3&spage=211",
  abstract =     "This paper sketches a technique for improving the rate
                 of convergence of a general oscillatory sequence, and
                 then applies this series acceleration algorithm to the
                 polylogarithm and the Hurwitz zeta function. As such,
                 it may be taken as an extension of the techniques given
                 by Borwein's ``An efficient algorithm for computing the
                 Riemann zeta function'' by Borwein for computing the
                 Riemann zeta function, to more general series. The
                 algorithm provides a rapid means of evaluating $
                 \operatorname {Li}_s(z) $ for general values of complex
                 $s$ and a kidney-shaped region of complex $z$ values
                 given by $ |z^2 / (z - 1)| < 4 $. By using the
                 duplication formula and the inversion formula, the
                 range of convergence for the polylogarithm may be
                 extended to the entire complex $z$-plane, and so the
                 algorithms described here allow for the evaluation of
                 the polylogarithm for all complex $s$ and $z$ values.
                 Alternatively, the Hurwitz zeta can be very rapidly
                 evaluated by means of an Euler Maclaurin series. The
                 polylogarithm and the Hurwitz zeta are related, in that
                 two evaluations of the one can be used to obtain a
                 value of the other; thus, either algorithm can be used
                 to evaluate either function. The Euler Maclaurin series
                 is a clear performance winner for the Hurwitz zeta,
                 while the Borwein algorithm is superior for evaluating
                 the polylogarithm in the kidney-shaped region. Both
                 algorithms are superior to the simple Taylor's series
                 or direct summation. The primary, concrete result of
                 this paper is an algorithm allows the exploration of
                 the Hurwitz zeta in the critical strip, where fast
                 algorithms are otherwise unavailable. A discussion of
                 the monodromy group of the polylogarithm is included.",
  acknowledgement = ack-nhfb,
  fjournal =     "Numerical Algorithms",
  journal-URL =  "http://link.springer.com/journal/11075",
  keywords =     "convergence acceleration",
}


bibtype: book
filename: elefunt.bib
label: Ware:2008:RIE
author: Willis H. Ware
title: RAND and the information evolution: a history in essays and vignettes
publisher: Rand Corporation
address: Santa Monica, CA
pages: xxvi + 201
year: 2008
DOI: https://doi.org/10.7249/cp537rc
ISBN: 0-8330-4513-X, 0-8330-4816-3, 1-282-45123-5 OR 083304513X, 0833048163, 1282451235
LCCN: QA76.27
bibdate: Tue Jun 2 19:14:18 MDT 2020
bibtimestamp: 2020.06.02 19:14:18 MDT
keywords: JOHNNIAC; JOSS; JOSS-1; JOSS-2; RAND tablet
acknowledgement: ack-nhfb
rawauthor: Willis H. Ware
rawtitle: {RAND} and the information evolution: a history in essays and vignettes
ISBN13:
978-0-8330-4513-3, 978-0-8330-4816-5, 978-1-282-45123-0 OR 9780833045133, 9780833048165, 9781282451230

bibsource:
fsz3950.oclc.org:210/WorldCat; https://www.math.utah.edu/pub/bibnet/authors/v/von-neumann-john.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/prng.bib; https://www.math.utah.edu/pub/tex/bib/unix.bib

URL:
http://www.jstor.org/stable/10.7249/cp537rc; https://www.rand.org/content/dam/rand/pubs/corporate_pubs/2008/RAND_CP537.pdf

abstract:
This professional memoir describes RAND's contributions to the evolution of computer science, particularly during the first decades following World War II, when digital computers succeeded slide rules, mechanical desk calculators, electric accounting machines, and analog computers. The memoir includes photographs and vignettes that reveal the collegial, creative, and often playful spirit in which the groundbreaking research was conducted at RAND.

TOC:
Dedication / v Preface / vii Figures / xiii Photographs / xv Tables / xvii Acknowledgments / xix Abbreviations / xxiii CHAPTER ONE Introduction / 1 Purpose and Scope / 1 Organization of the Document / 3 CHAPTER TWO The Department / 5 The Genesis of RAND / 5 The Need for a New Kind of Organization / 6 The Douglas Years / 7 An Independent, Private Nonprofit Organization / 8 The Nature of RAND's Contributions / 9 RAND Contributions to the Development of Computing / 10 In the Beginning / 10 An Early Computing Success / 11 The Move to Electronic Machines / 11 The Middle Years / 14 The JOHNNIAC Open-Shop System / 15 The Tablet / 16 Videographic System / 16 The Later Years / 17 RAND and the USAF Computing Evolution / 18 The Bottom Line / 19 CHAPTER THREE RAND's First Computer People / 21 The Legacy of Wartime Collaboration / 21 Early RAND Leaders / 22 Early Technical Staff / 24 The Douglas Thread / 24 The Wartime Thread / 26 The University Thread / 28 The Recruiting Thread / 30 Departmental Growth / 36 CHAPTER FOUR RAND's Early Computers / 45 Mid-20th Century Computation / 45 Reeves Electronic Analog Computer / 47 Plug-Board Interconnections / 50 Chopper-Stabilized Amplifiers / 50 Arbitrary Function Input / 51 The JOHNNIAC Digital Computer / 53 JOHNNIAC's ``Obituary'' / 63 IBM Mainframes / 64 Other Machinery. / 66 CHAPTER FIVE A Building for People with Computers / 67 A New Building and Campus. / 68 The Machine Room. / 72 Two-Story Installation / 72 REAC Installation. / 73 Raised-Floor Installation / 73 Air Conditioning. / 74 Configurations of the Machine Room / 75 Open House. / 75 Later Enhancements / 79 The Camera / 79 Kevershan's Trough / 80 Programmer-Alert Lights / 80 CHAPTER SIX Project Essays / 83 Approximations / 83 Random Digits and Normal Deviates / 87 The Bombing Simulator (aka Pinball Machine) / 90 The Air-Combat Room / 94 System Research Laboratory / 94 The RAND Tablet, Videographics, and Related Projects / 98 The RAND Tablet / 98 Handwriting Recognition / 99 Chinese-Character Lookup / 100 Map Annotation / 100 Videographic System / 103 GRAIL / 105 BIOMOD / 105 CLINFO / 107 Time-Shared Computing: JOSS / 109 JOSS-1 / 110 JOSS-2 / 113 Networked Computing: Packet Switching and Distributed Communications / 115 The Beginnings of Packet Switching: Some Underlying Concepts / 116 Text Editors (NED and e) / 122 Word Processing / 126 The Mail Handler / 128 The Original MH-Proposal Memorandum / 129 Implementation / 132 Another Perspective / 134 A User's Perspective / 135 The Developers' Present Views / 137 Artificial-Intelligence Research / 138 The Beginnings of Artificial Intelligence / 138 Newell, Shaw, and Simon: The Development of List-Processing Languages / 138 Expert Systems / 140 Knowledge-Based Simulation / 142 Computational Linguistics / 143 The Perfect Buddy / 144 Department of Defense Computer Institute / 147 Officer Career Paths / 149 Software / 150 Security and Privacy / 152 Security / 152 Privacy / 154 Fair Information Practices / 155 CHAPTER SEVEN Lore, Snippets, and Snapshots / 159 The Great Machine Fire / 159 The Gavel Caper / 159 Department-Head-Office Decor / 161 Oliver Alfred Gross and JOSS-1 / 162 The Soviet ``Threat'' / 163 Social Events / 164 The One-Way Wire / 166 Soviet Cybernetics / 166 Inter/Exhume / 167 The RAND Computer Symposia / 168 Professional Societies / 169 Microvignettes / 170 The Marchant March / 170 Getting Out the Documents / 171 Hero of the Week / 171 The Chiquita Banana War / 171 The Mengel Joint / 171 John Williams' Jaguar / 172 Programmer Sweepstakes / 173 CHAPTER EIGHT Epilogue / 175 Bibliography / 177 Index / 191

rawabstract:
This professional memoir describes RAND's contributions to the evolution of computer science, particularly during the first decades following World War II, when digital computers succeeded slide rules, mechanical desk calculators, electric accounting machines, and analog computers. The memoir includes photographs and vignettes that reveal the collegial, creative, and often playful spirit in which the groundbreaking research was conducted at RAND.

shortTOC:
Introduction The department RAND's first computer people RAND's early computers A building for people with computers Project essays Lore, snippets, and snapshots Epilogue

rawshortTOC:
Introduction \\ The department \\ RAND's first computer people \\ RAND's early computers \\ A building for people with computers \\ Project essays \\ Lore, snippets, and snapshots \\ Epilogue

rawTOC:
Dedication / v \\ Preface / vii \\ Figures / xiii \\ Photographs / xv \\ Tables / xvii \\ Acknowledgments / xix \\ Abbreviations / xxiii \\ CHAPTER ONE \\ Introduction / 1 \\ Purpose and Scope / 1 \\ Organization of the Document / 3 \\ CHAPTER TWO \\ The Department / 5 \\ The Genesis of RAND / 5 \\ The Need for a New Kind of Organization / 6 \\ The Douglas Years / 7 \\ An Independent, Private Nonprofit Organization / 8 \\ The Nature of RAND's Contributions / 9 \\ RAND Contributions to the Development of Computing / 10 \\ In the Beginning / 10 \\ An Early Computing Success / 11 \\ The Move to Electronic Machines / 11 \\ The Middle Years / 14 \\ The JOHNNIAC Open-Shop System / 15 \\ The Tablet / 16 \\ Videographic System / 16 \\ The Later Years / 17 \\ RAND and the USAF Computing Evolution / 18 \\ The Bottom Line / 19 \\ CHAPTER THREE \\ RAND's First Computer People / 21 \\ The Legacy of Wartime Collaboration / 21 \\ Early RAND Leaders / 22 \\ Early Technical Staff / 24 \\ The Douglas Thread / 24 \\ The Wartime Thread / 26 \\ The University Thread / 28 \\ The Recruiting Thread / 30 \\ Departmental Growth / 36 \\ CHAPTER FOUR \\ RAND's Early Computers / 45 \\ Mid-20th Century Computation / 45 \\ Reeves Electronic Analog Computer / 47 \\ Plug-Board Interconnections / 50 \\ Chopper-Stabilized Amplifiers / 50 \\ Arbitrary Function Input / 51 \\ The JOHNNIAC Digital Computer / 53 \\ JOHNNIAC's ``Obituary'' / 63 \\ IBM Mainframes / 64 \\ Other Machinery. / 66 \\ CHAPTER FIVE \\ A Building for People with Computers / 67 \\ A New Building and Campus. / 68 \\ The Machine Room. / 72 \\ Two-Story Installation / 72 \\ REAC Installation. / 73 \\ Raised-Floor Installation / 73 \\ Air Conditioning. / 74 \\ Configurations of the Machine Room / 75 \\ Open House. / 75 \\ Later Enhancements / 79 \\ The Camera / 79 \\ Kevershan's Trough / 80 \\ Programmer-Alert Lights / 80 \\ CHAPTER SIX \\ Project Essays / 83 \\ Approximations / 83 \\ Random Digits and Normal Deviates / 87 \\ The Bombing Simulator (aka Pinball Machine) / 90 \\ The Air-Combat Room / 94 \\ System Research Laboratory / 94 \\ The RAND Tablet, Videographics, and Related Projects / 98 \\ The RAND Tablet / 98 \\ Handwriting Recognition / 99 \\ Chinese-Character Lookup / 100 \\ Map Annotation / 100 \\ Videographic System / 103 \\ GRAIL / 105 \\ BIOMOD / 105 \\ CLINFO / 107 \\ Time-Shared Computing: JOSS / 109 \\ JOSS-1 / 110 \\ JOSS-2 / 113 \\ Networked Computing: Packet Switching and Distributed Communications / 115 \\ The Beginnings of Packet Switching: Some Underlying Concepts / 116 \\ Text Editors (NED and e) / 122 \\ Word Processing / 126 \\ The Mail Handler / 128 \\ The Original MH-Proposal Memorandum / 129 \\ Implementation / 132 \\ Another Perspective / 134 \\ A User's Perspective / 135 \\ The Developers' Present Views / 137 \\ Artificial-Intelligence Research / 138 \\ The Beginnings of Artificial Intelligence / 138 \\ Newell, Shaw, and Simon: The Development of List-Processing Languages / 138 \\ Expert Systems / 140 \\ Knowledge-Based Simulation / 142 \\ Computational Linguistics / 143 \\ The Perfect Buddy / 144 \\ Department of Defense Computer Institute / 147 \\ Officer Career Paths / 149 \\ Software / 150 \\ Security and Privacy / 152 \\ Security / 152 \\ Privacy / 154 \\ Fair Information Practices / 155 \\ CHAPTER SEVEN \\ Lore, Snippets, and Snapshots / 159 \\ The Great Machine Fire / 159 \\ The Gavel Caper / 159 \\ Department-Head-Office Decor / 161 \\ Oliver Alfred Gross and JOSS-1 / 162 \\ The Soviet ``Threat'' / 163 \\ Social Events / 164 \\ The One-Way Wire / 166 \\ Soviet Cybernetics / 166 \\ Inter/Exhume / 167 \\ The RAND Computer Symposia / 168 \\ Professional Societies / 169 \\ Microvignettes / 170 \\ The Marchant March / 170 \\ Getting Out the Documents / 171 \\ Hero of the Week / 171 \\ The Chiquita Banana War / 171 \\ The Mengel Joint / 171 \\ John Williams' Jaguar / 172 \\ Programmer Sweepstakes / 173 \\ CHAPTER EIGHT \\ Epilogue / 175 \\ Bibliography / 177 \\ Index / 191

entry:
@Book{Ware:2008:RIE,
  author =       "Willis H. Ware",
  title =        "{RAND} and the information evolution: a history in
                 essays and vignettes",
  publisher =    "Rand Corporation",
  address =      "Santa Monica, CA",
  pages =        "xxvi + 201",
  year =         "2008",
  DOI =          "https://doi.org/10.7249/cp537rc",
  ISBN =         "0-8330-4513-X, 0-8330-4816-3, 1-282-45123-5",
  ISBN-13 =      "978-0-8330-4513-3, 978-0-8330-4816-5,
                 978-1-282-45123-0",
  LCCN =         "QA76.27",
  bibdate =      "Tue Jun 2 19:14:18 MDT 2020",
  bibsource =    "fsz3950.oclc.org:210/WorldCat;
                 https://www.math.utah.edu/pub/bibnet/authors/v/von-neumann-john.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/prng.bib;
                 https://www.math.utah.edu/pub/tex/bib/unix.bib",
  URL =          "http://www.jstor.org/stable/10.7249/cp537rc;
                 https://www.rand.org/content/dam/rand/pubs/corporate_pubs/2008/RAND_CP537.pdf",
  abstract =     "This professional memoir describes RAND's
                 contributions to the evolution of computer science,
                 particularly during the first decades following World
                 War II, when digital computers succeeded slide rules,
                 mechanical desk calculators, electric accounting
                 machines, and analog computers. The memoir includes
                 photographs and vignettes that reveal the collegial,
                 creative, and often playful spirit in which the
                 groundbreaking research was conducted at RAND.",
  acknowledgement = ack-nhfb,
  keywords =     "JOHNNIAC; JOSS; JOSS-1; JOSS-2; RAND tablet",
  remark-1 =     "Page 13 has a photograph of the JOHNNIAC, and on the
                 wall of its room, a photograph of John von Neumann.",
  remark-2 =     "From page 15: ``\ldots{} the JOHNNIAC, which
                 nonetheless was the basis of a continuing series of
                 engineering advances, each making important
                 contributions to the art of the time. Among them were
                 the first commercially produced magnetic core memory,
                 which, for a while, was the largest in existence [4096
                 40-bit words]; a transistor-based adder and logic which
                 caused the JOHNNIAC to become a hybrid
                 transistor-vacuum tube device; the first high-speed
                 impact printer 140 columns wide (manufactured by
                 Anderson--Nichols, an engineering contracting firm);
                 and the first machine with extensive trouble-diagnostic
                 capability from the operating console.''",
  remark-3 =     "From page 53: ``the only bright spot was the Princeton
                 development at IAS, and thus it was that a working
                 alliance between RAND and IAS came into being. RAND
                 would build a machine patterned in the likeness of the
                 Princeton one. So JOHNNIAC came from an illustrious
                 ancestor --- the so-called von Neumann machine
                 developed at Princeton's IAS.''",
  remark-4 =     "Page 57 has a photograph of the JOHNNIAC's 256-word
                 Selectron high-speed memory. Page 59, a picture of its
                 140-column drum printer. Page 61 has an inside view of
                 the JOHNNIAC. Page 73 shows a step in the installation
                 of the JOHNNIAC. Page 162 has a photograph of the
                 JOHNNIAC console.",
  remark-5 =     "From page 66: ``RAND purchased the first commercially
                 available license for UNIX.''",
  remark-6 =     "Page 84 has a photo of a young Cecil Hastings, an
                 early pioneer of function approximation on digital
                 computers, and a few paragraphs about his work and its
                 influence.",
  remark-7 =     "Pages 87--90 discuss the preparation of RAND's famous
                 book of one million random digits, computed in Spring
                 1947, tested for two years after that before
                 publication in 1955. About 7000 copies of the book were
                 sold over three printings and fifteen years, and the
                 book was reprinted in 1966 and 2001.",
  remark-8 =     "From page 138: ``In the 1950s, RAND was involved in
                 designing and building one of the first stored-program
                 digital computers, the JOHNNIAC (named after John von
                 Neumann, a RAND consultant in the late 1940s and early
                 1950s). It was in operation from 1953 to 1966,
                 \ldots{}.''",
  shorttableofcontents = "Introduction \\
                 The department \\
                 RAND's first computer people \\
                 RAND's early computers \\
                 A building for people with computers \\
                 Project essays \\
                 Lore, snippets, and snapshots \\
                 Epilogue",
  tableofcontents = "Dedication / v \\
                 Preface / vii \\
                 Figures / xiii \\
                 Photographs / xv \\
                 Tables / xvii \\
                 Acknowledgments / xix \\
                 Abbreviations / xxiii \\
                 CHAPTER ONE \\
                 Introduction / 1 \\
                 Purpose and Scope / 1 \\
                 Organization of the Document / 3 \\
                 CHAPTER TWO \\
                 The Department / 5 \\
                 The Genesis of RAND / 5 \\
                 The Need for a New Kind of Organization / 6 \\
                 The Douglas Years / 7 \\
                 An Independent, Private Nonprofit Organization / 8 \\
                 The Nature of RAND's Contributions / 9 \\
                 RAND Contributions to the Development of Computing / 10
                 \\
                 In the Beginning / 10 \\
                 An Early Computing Success / 11 \\
                 The Move to Electronic Machines / 11 \\
                 The Middle Years / 14 \\
                 The JOHNNIAC Open-Shop System / 15 \\
                 The Tablet / 16 \\
                 Videographic System / 16 \\
                 The Later Years / 17 \\
                 RAND and the USAF Computing Evolution / 18 \\
                 The Bottom Line / 19 \\
                 CHAPTER THREE \\
                 RAND's First Computer People / 21 \\
                 The Legacy of Wartime Collaboration / 21 \\
                 Early RAND Leaders / 22 \\
                 Early Technical Staff / 24 \\
                 The Douglas Thread / 24 \\
                 The Wartime Thread / 26 \\
                 The University Thread / 28 \\
                 The Recruiting Thread / 30 \\
                 Departmental Growth / 36 \\
                 CHAPTER FOUR \\
                 RAND's Early Computers / 45 \\
                 Mid-20th Century Computation / 45 \\
                 Reeves Electronic Analog Computer / 47 \\
                 Plug-Board Interconnections / 50 \\
                 Chopper-Stabilized Amplifiers / 50 \\
                 Arbitrary Function Input / 51 \\
                 The JOHNNIAC Digital Computer / 53 \\
                 JOHNNIAC's ``Obituary'' / 63 \\
                 IBM Mainframes / 64 \\
                 Other Machinery. / 66 \\
                 CHAPTER FIVE \\
                 A Building for People with Computers / 67 \\
                 A New Building and Campus. / 68 \\
                 The Machine Room. / 72 \\
                 Two-Story Installation / 72 \\
                 REAC Installation. / 73 \\
                 Raised-Floor Installation / 73 \\
                 Air Conditioning. / 74 \\
                 Configurations of the Machine Room / 75 \\
                 Open House. / 75 \\
                 Later Enhancements / 79 \\
                 The Camera / 79 \\
                 Kevershan's Trough / 80 \\
                 Programmer-Alert Lights / 80 \\
                 CHAPTER SIX \\
                 Project Essays / 83 \\
                 Approximations / 83 \\
                 Random Digits and Normal Deviates / 87 \\
                 The Bombing Simulator (aka Pinball Machine) / 90 \\
                 The Air-Combat Room / 94 \\
                 System Research Laboratory / 94 \\
                 The RAND Tablet, Videographics, and Related Projects /
                 98 \\
                 The RAND Tablet / 98 \\
                 Handwriting Recognition / 99 \\
                 Chinese-Character Lookup / 100 \\
                 Map Annotation / 100 \\
                 Videographic System / 103 \\
                 GRAIL / 105 \\
                 BIOMOD / 105 \\
                 CLINFO / 107 \\
                 Time-Shared Computing: JOSS / 109 \\
                 JOSS-1 / 110 \\
                 JOSS-2 / 113 \\
                 Networked Computing: Packet Switching and Distributed
                 Communications / 115 \\
                 The Beginnings of Packet Switching: Some Underlying
                 Concepts / 116 \\
                 Text Editors (NED and e) / 122 \\
                 Word Processing / 126 \\
                 The Mail Handler / 128 \\
                 The Original MH-Proposal Memorandum / 129 \\
                 Implementation / 132 \\
                 Another Perspective / 134 \\
                 A User's Perspective / 135 \\
                 The Developers' Present Views / 137 \\
                 Artificial-Intelligence Research / 138 \\
                 The Beginnings of Artificial Intelligence / 138 \\
                 Newell, Shaw, and Simon: The Development of
                 List-Processing Languages / 138 \\
                 Expert Systems / 140 \\
                 Knowledge-Based Simulation / 142 \\
                 Computational Linguistics / 143 \\
                 The Perfect Buddy / 144 \\
                 Department of Defense Computer Institute / 147 \\
                 Officer Career Paths / 149 \\
                 Software / 150 \\
                 Security and Privacy / 152 \\
                 Security / 152 \\
                 Privacy / 154 \\
                 Fair Information Practices / 155 \\
                 CHAPTER SEVEN \\
                 Lore, Snippets, and Snapshots / 159 \\
                 The Great Machine Fire / 159 \\
                 The Gavel Caper / 159 \\
                 Department-Head-Office Decor / 161 \\
                 Oliver Alfred Gross and JOSS-1 / 162 \\
                 The Soviet ``Threat'' / 163 \\
                 Social Events / 164 \\
                 The One-Way Wire / 166 \\
                 Soviet Cybernetics / 166 \\
                 Inter/Exhume / 167 \\
                 The RAND Computer Symposia / 168 \\
                 Professional Societies / 169 \\
                 Microvignettes / 170 \\
                 The Marchant March / 170 \\
                 Getting Out the Documents / 171 \\
                 Hero of the Week / 171 \\
                 The Chiquita Banana War / 171 \\
                 The Mengel Joint / 171 \\
                 John Williams' Jaguar / 172 \\
                 Programmer Sweepstakes / 173 \\
                 CHAPTER EIGHT \\
                 Epilogue / 175 \\
                 Bibliography / 177 \\
                 Index / 191",
}


bibtype: article
filename: elefunt.bib
label: Zhu:2008:SNR
author: Ling Zhu and Jinju Sun
title: Six new Redheffer-type inequalities for circular and hyperbolic functions
journal: j-COMPUT-MATH-APPL
volume: 56
number: 2
pages: 522--529
month: jul
monthnumber: 7
year: 2008
CODEN: CMAPDK
ISSN: 0898-1221 (print), 1873-7668 (electronic) OR 08981221 (print), 18737668 (electronic)
ISSNL: 0898-1221 OR 08981221
bibdate: Wed Mar 1 21:50:15 MST 2017
bibtimestamp: 2017.03.01 21:50:15 MST
URL: http://www.sciencedirect.com/science/article/pii/S0898122108000813
fjournal: Computers and Mathematics with Applications
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/08981221
rawauthor: Ling Zhu and Jinju Sun
rawtitle: Six new {Redheffer}-type inequalities for circular and hyperbolic functions
entry:
@Article{Zhu:2008:SNR,
  author =       "Ling Zhu and Jinju Sun",
  title =        "Six new {Redheffer}-type inequalities for circular and
                 hyperbolic functions",
  journal =      j-COMPUT-MATH-APPL,
  volume =       "56",
  number =       "2",
  pages =        "522--529",
  month =        jul,
  year =         "2008",
  CODEN =        "CMAPDK",
  ISSN =         "0898-1221 (print), 1873-7668 (electronic)",
  ISSN-L =       "0898-1221",
  bibdate =      "Wed Mar 1 21:50:15 MST 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/computmathappl2000.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0898122108000813",
  acknowledgement = ack-nhfb,
  fjournal =     "Computers and Mathematics with Applications",
  journal-URL =  "http://www.sciencedirect.com/science/journal/08981221",
}


bibtype: article
filename: elefunt.bib
label: Anand:2009:OCS
author: C. K. Anand and W. Kahl
title: An Optimized Cell BE Special Function Library Generated by Coconut
journal: j-IEEE-TRANS-COMPUT
volume: 58
number: 8
pages: 1126--1138
month: aug
monthnumber: 8
year: 2009
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2008.223
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Mon Jul 4 11:37:43 MDT 2011
bibtimestamp: 2011.07.04 11:37:43 MDT
URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4731241
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: C. K. Anand and W. Kahl
rawtitle: An Optimized {Cell BE} Special Function Library Generated by {Coconut}
abstract:
Coconut, a tool for developing high-assurance, high-performance kernels for scientific computing, contains an extensible domain-specific language (DSL) embedded in Haskell. The DSL supports interactive prototyping and unit testing, simplifying the process of designing efficient implementations of common patterns. Unscheduled C and scheduled assembly language output are supported. Using the patterns, even nonexpert users can write efficient function implementations, leveraging special hardware features. A production-quality library of elementary functions for the cell BE SPU compute engines has been developed. Coconut-generated and -scheduled vector functions were more than four times faster than commercially distributed functions written in C with intrinsics (a nicer syntax for in-line assembly), wrapped in loops and scheduled by tt spuxlc. All Coconut functions were faster, but the difference was larger for hard-to-approximate functions for which register-level SIMD lookups made a bigger difference. Other helpful features in the language include facilities for translating interval and polynomial descriptions between GHCi, a Haskell interpreter used to prototype in the DSL, and Maple, used for exploration and minimax polynomial generation. This makes it easier to match mathematical properties of the functions with efficient calculational patterns in the SPU ISA. By using single, literate source files, the resulting functions are remarkably readable.

rawabstract:
Coconut, a tool for developing high-assurance, high-performance kernels for scientific computing, contains an extensible domain-specific language (DSL) embedded in Haskell. The DSL supports interactive prototyping and unit testing, simplifying the process of designing efficient implementations of common patterns. Unscheduled C and scheduled assembly language output are supported. Using the patterns, even nonexpert users can write efficient function implementations, leveraging special hardware features. A production-quality library of elementary functions for the cell BE SPU compute engines has been developed. Coconut-generated and -scheduled vector functions were more than four times faster than commercially distributed functions written in C with intrinsics (a nicer syntax for in-line assembly), wrapped in loops and scheduled by {\tt spuxlc}. All Coconut functions were faster, but the difference was larger for hard-to-approximate functions for which register-level SIMD lookups made a bigger difference. Other helpful features in the language include facilities for translating interval and polynomial descriptions between GHCi, a Haskell interpreter used to prototype in the DSL, and Maple, used for exploration and minimax polynomial generation. This makes it easier to match mathematical properties of the functions with efficient calculational patterns in the SPU ISA. By using single, literate source files, the resulting functions are remarkably readable.

entry:
@Article{Anand:2009:OCS,
  author =       "C. K. Anand and W. Kahl",
  title =        "An Optimized {Cell BE} Special Function Library
                 Generated by {Coconut}",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "58",
  number =       "8",
  pages =        "1126--1138",
  month =        aug,
  year =         "2009",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2008.223",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Mon Jul 4 11:37:43 MDT 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib",
  URL =          "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4731241",
  abstract =     "Coconut, a tool for developing high-assurance,
                 high-performance kernels for scientific computing,
                 contains an extensible domain-specific language (DSL)
                 embedded in Haskell. The DSL supports interactive
                 prototyping and unit testing, simplifying the process
                 of designing efficient implementations of common
                 patterns. Unscheduled C and scheduled assembly language
                 output are supported. Using the patterns, even
                 nonexpert users can write efficient function
                 implementations, leveraging special hardware features.
                 A production-quality library of elementary functions
                 for the cell BE SPU compute engines has been developed.
                 Coconut-generated and -scheduled vector functions were
                 more than four times faster than commercially
                 distributed functions written in C with intrinsics (a
                 nicer syntax for in-line assembly), wrapped in loops
                 and scheduled by {\tt spuxlc}. All Coconut functions
                 were faster, but the difference was larger for
                 hard-to-approximate functions for which register-level
                 SIMD lookups made a bigger difference. Other helpful
                 features in the language include facilities for
                 translating interval and polynomial descriptions
                 between GHCi, a Haskell interpreter used to prototype
                 in the DSL, and Maple, used for exploration and minimax
                 polynomial generation. This makes it easier to match
                 mathematical properties of the functions with efficient
                 calculational patterns in the SPU ISA. By using single,
                 literate source files, the resulting functions are
                 remarkably readable.",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
}


bibtype: article
filename: elefunt.bib
label: Backeljauw:2009:ACF
author: Franky Backeljauw and Annie Cuyt
title: Algorithm 895: a continued fractions package for special functions
journal: j-TOMS
volume: 36
number: 3
pages: 15:1--15:20
month: jul
monthnumber: 7
year: 2009
CODEN: ACMSCU
DOI: https://doi.org/10.1145/1527286.1527289
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
bibdate: Tue Jul 21 14:09:07 MDT 2009
bibtimestamp: 2009.07.21 14:09:07 MDT
fjournal: ACM Transactions on Mathematical Software (TOMS)
keywords: CAS software; continued fractions; Maple; special functions
acknowledgement: ack-nhfb
articleno: 15
journalURL: https://dl.acm.org/loi/toms
rawauthor: Franky Backeljauw and Annie Cuyt
rawtitle: {Algorithm 895}: a continued fractions package for special functions
abstract:
The continued fractions for special functions package (in the sequel abbreviated as CFSF package) complements a systematic study of continued fraction representations for special functions. It provides all the functionality to create continued fractions, in particular $k$-periodic or limit $k$-periodic fractions, to compute approximants, make use of continued fraction tails, perform equivalence transformations and contractions, and much more. The package, developed in Maple, includes a library of more than 200 representations of special functions, of which only 10% can be found in the 1964 NBS em Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables by M. Abramowitz and I. Stegun.

rawabstract:
The continued fractions for special functions package (in the sequel abbreviated as CFSF package) complements a systematic study of continued fraction representations for special functions. It provides all the functionality to create continued fractions, in particular $k$-periodic or limit $k$-periodic fractions, to compute approximants, make use of continued fraction tails, perform equivalence transformations and contractions, and much more. The package, developed in Maple, includes a library of more than 200 representations of special functions, of which only 10\% can be found in the 1964 NBS {\em Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables\/} by M. Abramowitz and I. Stegun.

entry:
@Article{Backeljauw:2009:ACF,
  author =       "Franky Backeljauw and Annie Cuyt",
  title =        "{Algorithm 895}: a continued fractions package for
                 special functions",
  journal =      j-TOMS,
  volume =       "36",
  number =       "3",
  pages =        "15:1--15:20",
  month =        jul,
  year =         "2009",
  CODEN =        "ACMSCU",
  DOI =          "https://doi.org/10.1145/1527286.1527289",
  ISSN =         "0098-3500 (print), 1557-7295 (electronic)",
  ISSN-L =       "0098-3500",
  bibdate =      "Tue Jul 21 14:09:07 MDT 2009",
  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 =     "The continued fractions for special functions package
                 (in the sequel abbreviated as CFSF package) complements
                 a systematic study of continued fraction
                 representations for special functions. It provides all
                 the functionality to create continued fractions, in
                 particular $k$-periodic or limit $k$-periodic
                 fractions, to compute approximants, make use of
                 continued fraction tails, perform equivalence
                 transformations and contractions, and much more. The
                 package, developed in Maple, includes a library of more
                 than 200 representations of special functions, of which
                 only 10\% can be found in the 1964 NBS {\em Handbook of
                 Mathematical Functions with Formulas, Graphs and
                 Mathematical Tables\/} by M. Abramowitz and I.
                 Stegun.",
  acknowledgement = ack-nhfb,
  articleno =    "15",
  fjournal =     "ACM Transactions on Mathematical Software (TOMS)",
  journal-URL =  "https://dl.acm.org/loi/toms",
  keywords =     "CAS software; continued fractions; Maple; special
                 functions",
}


bibtype: article
filename: elefunt.bib
label: Blomquist:2009:MSC
author: Frithjof Blomquist and Werner Hofschuster and Walter Kramer
title: A Modified Staggered Correction Arithmetic with Enhanced Accuracy and Very Wide Exponent Range
journal: j-LECT-NOTES-COMP-SCI
volume: 5492
pages: 41--67
year: 2009
CODEN: LNCSD9
DOI: https://doi.org/10.1007/978-3-642-01591-5_4
ISSN: 0302-9743 (print), 1611-3349 (electronic) OR 03029743 (print), 16113349 (electronic)
ISSNL: 0302-9743 OR 03029743
bibdate: Tue Apr 10 08:32:19 2012
bibtimestamp: 2012.04.10 08:32:19 ???
URL: http://www.springerlink.com/content/k038294004403504/
fjournal: Lecture Notes in Computer Science
remark: Conference on Numerical Validation in Current Hardware Architectures
acknowledgement: ack-nhfb
authordates: 1952--2014 (WK)
journalURL: http://link.springer.com/bookseries/558
rawauthor: Frithjof Blomquist and Werner Hofschuster and Walter Kr{\"a}mer
rawtitle: A Modified Staggered Correction Arithmetic with Enhanced Accuracy and Very Wide Exponent Range
keywords:
C-XSC; complex interval functions; interval computation; multiple precision; reliable numerical computations; staggered correction; wide exponent range

entry:
@Article{Blomquist:2009:MSC,
  author =       "Frithjof Blomquist and Werner Hofschuster and Walter
                 Kr{\"a}mer",
  title =        "A Modified Staggered Correction Arithmetic with
                 Enhanced Accuracy and Very Wide Exponent Range",
  journal =      j-LECT-NOTES-COMP-SCI,
  volume =       "5492",
  pages =        "41--67",
  year =         "2009",
  CODEN =        "LNCSD9",
  DOI =          "https://doi.org/10.1007/978-3-642-01591-5_4",
  ISSN =         "0302-9743 (print), 1611-3349 (electronic)",
  ISSN-L =       "0302-9743",
  bibdate =      "Tue Apr 10 08:32:19 2012",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  URL =          "http://www.springerlink.com/content/k038294004403504/",
  acknowledgement = ack-nhfb,
  author-dates = "1952--2014 (WK)",
  fjournal =     "Lecture Notes in Computer Science",
  journal-URL =  "http://link.springer.com/bookseries/558",
  keywords =     "C-XSC; complex interval functions; interval
                 computation; multiple precision; reliable numerical
                 computations; staggered correction; wide exponent
                 range",
  remark =       "Conference on Numerical Validation in Current Hardware
                 Architectures",
  remark-2 =     "Includes algorithms for division, $\exp(x)$, $(1 +
                 x)^n$, $\log(x)$, $\log(1 + x)$, and $\sqrt{x}$.
                 Staggered arithmetic represents numbers with tuples
                 $(e, x_1, x_2, \ldots{}, x_n)$ where $e$ is either
                 integer or a floating-point whole number, the $x_k$ are
                 floating-point, and a number has the value $2^e \sum_{k
                 = 1}^n x_k$. For interval arithmetic, the last element
                 is a pair of lower and upper bounds.",
}


bibtype: article
filename: elefunt.bib
label: Boldo:2009:FVA
author: S. Boldo and M. Daumas and Ren-Cang Li
title: Formally Verified Argument Reduction with a Fused Multiply-Add
journal: j-IEEE-TRANS-COMPUT
volume: 58
number: 8
pages: 1139--1145
month: aug
monthnumber: 8
year: 2009
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2008.216
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Mon Jul 4 11:37:43 MDT 2011
bibtimestamp: 2011.07.04 11:37:43 MDT
URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4711042
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: S. Boldo and M. Daumas and Ren-Cang Li
rawtitle: Formally Verified Argument Reduction with a Fused Multiply-Add
abstract:
The Cody and Waite argument reduction technique works perfectly for reasonably large arguments, but as the input grows, there are no bits left to approximate the constant with enough accuracy. Under mild assumptions, we show that the result computed with a fused multiply-add provides a fully accurate result for many possible values of the input with a constant almost accurate to the full working precision. We also present an algorithm for a fully accurate second reduction step to reach full double accuracy (all the significand bits of two numbers are accurate) even in the worst cases of argument reduction. Our work recalls the common algorithms and presents proofs of correctness. All the proofs are formally verified using the Coq automatic proof checker.

rawabstract:
The Cody and Waite argument reduction technique works perfectly for reasonably large arguments, but as the input grows, there are no bits left to approximate the constant with enough accuracy. Under mild assumptions, we show that the result computed with a fused multiply-add provides a fully accurate result for many possible values of the input with a constant almost accurate to the full working precision. We also present an algorithm for a fully accurate second reduction step to reach full double accuracy (all the significand bits of two numbers are accurate) even in the worst cases of argument reduction. Our work recalls the common algorithms and presents proofs of correctness. All the proofs are formally verified using the Coq automatic proof checker.

entry:
@Article{Boldo:2009:FVA,
  author =       "S. Boldo and M. Daumas and Ren-Cang Li",
  title =        "Formally Verified Argument Reduction with a Fused
                 Multiply-Add",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "58",
  number =       "8",
  pages =        "1139--1145",
  month =        aug,
  year =         "2009",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2008.216",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Mon Jul 4 11:37:43 MDT 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib",
  URL =          "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4711042",
  abstract =     "The Cody and Waite argument reduction technique works
                 perfectly for reasonably large arguments, but as the
                 input grows, there are no bits left to approximate the
                 constant with enough accuracy. Under mild assumptions,
                 we show that the result computed with a fused
                 multiply-add provides a fully accurate result for many
                 possible values of the input with a constant almost
                 accurate to the full working precision. We also present
                 an algorithm for a fully accurate second reduction step
                 to reach full double accuracy (all the significand bits
                 of two numbers are accurate) even in the worst cases of
                 argument reduction. Our work recalls the common
                 algorithms and presents proofs of correctness. All the
                 proofs are formally verified using the Coq automatic
                 proof checker.",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
}


bibtype: article
filename: elefunt.bib
label: Bowling:2009:LAC
author: Shannon R. Bowling and Mohammad T. Khasawneh and Sittichai Kaewkuekool and Byung Rae Cho
title: A logistic approximation to the cumulative normal distribution
journal: Journal of Industrial Engineering and Management
volume: 2
number: 1
pages: 114--127
year: 2009
DOI: https://doi.org/10.3926/jiem..v2n1.p114-127
bibdate: Sat Dec 16 15:22:05 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2017.12.16 15:22:05 ???
URL: http://www.jiem.org/index.php/jiem/article/view/60
fjournal: Journal of Industrial Engineering and Management
acknowledgement: ack-nhfb
ajournal: J. Ind. Eng. Manage.
journalURL: http://www.jiem.org/index.php/jiem/
rawauthor: Shannon R. Bowling and Mohammad T. Khasawneh and Sittichai Kaewkuekool and Byung Rae Cho
rawtitle: A logistic approximation to the cumulative normal distribution
entry:
@Article{Bowling:2009:LAC,
  author =       "Shannon R. Bowling and Mohammad T. Khasawneh and
                 Sittichai Kaewkuekool and Byung Rae Cho",
  title =        "A logistic approximation to the cumulative normal
                 distribution",
  journal =      "Journal of Industrial Engineering and Management",
  volume =       "2",
  number =       "1",
  pages =        "114--127",
  month =        "",
  year =         "2009",
  DOI =          "https://doi.org/10.3926/jiem..v2n1.p114-127",
  bibdate =      "Sat Dec 16 15:22:05 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.jiem.org/index.php/jiem/article/view/60",
  acknowledgement = ack-nhfb,
  ajournal =     "J. Ind. Eng. Manage.",
  fjournal =     "Journal of Industrial Engineering and Management",
  journal-URL =  "http://www.jiem.org/index.php/jiem/",
}


bibtype: article
filename: elefunt.bib
label: Boyd:2009:AAC
author: John P. Boyd
journal: j-J-COMPUT-PHYS
volume: 228
number: 5
pages: 1404--1411
day: 20
month: mar
monthnumber: 3
year: 2009
CODEN: JCTPAH
DOI: https://doi.org/10.1016/j.jcp.2008.10.039
ISSN: 0021-9991 (print), 1090-2716 (electronic) OR 00219991 (print), 10902716 (electronic)
ISSNL: 0021-9991 OR 00219991
bibdate: Thu Dec 01 10:35:35 2011
bibtimestamp: 2011.12.01 10:35:35 ???
fjournal: Journal of Computational Physics
keywords: Bernoulli polynomials; Clausen functions; convergence acceleration; Lanczos--Krylov (LK) functions
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00219991/
rawauthor: John P. Boyd
title:
Acceleration of algebraically-converging Fourier series when the coefficients have series in powers of $ 1 / n $

bibsource:
https://www.math.utah.edu/pub/tex/bib/elefunt.bib; https://www.math.utah.edu/pub/tex/bib/jcomputphys2000.bib

rawtitle:
Acceleration of algebraically-converging {Fourier} series when the coefficients have series in powers of $ 1 / n $

entry:
@Article{Boyd:2009:AAC,
  author =       "John P. Boyd",
  title =        "Acceleration of algebraically-converging {Fourier}
                 series when the coefficients have series in powers of $
                 1 / n $",
  journal =      j-J-COMPUT-PHYS,
  volume =       "228",
  number =       "5",
  pages =        "1404--1411",
  day =          "20",
  month =        mar,
  year =         "2009",
  CODEN =        "JCTPAH",
  DOI =          "https://doi.org/10.1016/j.jcp.2008.10.039",
  ISSN =         "0021-9991 (print), 1090-2716 (electronic)",
  ISSN-L =       "0021-9991",
  bibdate =      "Thu Dec 01 10:35:35 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jcomputphys2000.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "Journal of Computational Physics",
  journal-URL =  "http://www.sciencedirect.com/science/journal/00219991/",
  keywords =     "Bernoulli polynomials; Clausen functions; convergence
                 acceleration; Lanczos--Krylov (LK) functions",
}


bibtype: article
filename: elefunt.bib
label: Bunck:2009:FAE
author: Benjamin F. Bunck
title: A fast algorithm for evaluation of normalized Hermite functions
journal: j-BIT-NUM-MATH
volume: 49
number: 2
pages: 281--295
month: jun
monthnumber: 6
year: 2009
CODEN: BITTEL, NBITAB
ISSN: 0006-3835 (print), 1572-9125 (electronic) OR 00063835 (print), 15729125 (electronic)
ISSNL: 0006-3835 OR 00063835
bibdate: Mon May 24 15:36:43 MDT 2010
bibtimestamp: 2010.05.24 15:36:43 MDT
URL: http://www.springerlink.com/openurl.asp?genre=article&issn=0006-3835&volume=49&issue=2&spage=281
fjournal: BIT. Numerical Mathematics
keywords: fast algorithm; Hermite functions; numerical integration; recursion
acknowledgement: ack-nhfb
journalURL: http://link.springer.com/journal/10543
rawauthor: Benjamin F. Bunck
rawtitle: A fast algorithm for evaluation of normalized {Hermite} functions
bibsource:
http://springerlink.metapress.com/openurl.asp?genre=issue&issn=0006-3835&volume=49&issue=2; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

abstract:
An algorithm for computing the normalized Hermite Functions, $ h_n(x) $, in floating point arithmetic is presented. The algorithm is based on an efficient numerical evaluation of certain closed contour integrals in the complex plane. For large degree $n$, the algorithm is significantly faster than the $ O(n) $ complexity of the well known three-term recurrence relation. Comparable accuracy is achieved in no more $ O(sqrt n) $ than operations, and for arguments bounded away from $ pm sqrt 2n $, only $ O(sqrt log n) $ operations.

rawabstract:
An algorithm for computing the normalized Hermite Functions, $ h_n(x) $, in floating point arithmetic is presented. The algorithm is based on an efficient numerical evaluation of certain closed contour integrals in the complex plane. For large degree $n$, the algorithm is significantly faster than the $ O(n) $ complexity of the well known three-term recurrence relation. Comparable accuracy is achieved in no more $ O(\sqrt {n}) $ than operations, and for arguments bounded away from $ \pm \sqrt {2n} $, only $ O(\sqrt {\log n}) $ operations.

entry:
@Article{Bunck:2009:FAE,
  author =       "Benjamin F. Bunck",
  title =        "A fast algorithm for evaluation of normalized
                 {Hermite} functions",
  journal =      j-BIT-NUM-MATH,
  volume =       "49",
  number =       "2",
  pages =        "281--295",
  month =        jun,
  year =         "2009",
  CODEN =        "BITTEL, NBITAB",
  ISSN =         "0006-3835 (print), 1572-9125 (electronic)",
  ISSN-L =       "0006-3835",
  bibdate =      "Mon May 24 15:36:43 MDT 2010",
  bibsource =    "http://springerlink.metapress.com/openurl.asp?genre=issue&issn=0006-3835&volume=49&issue=2;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.springerlink.com/openurl.asp?genre=article&issn=0006-3835&volume=49&issue=2&spage=281",
  abstract =     "An algorithm for computing the normalized Hermite
                 Functions, $ h_n(x) $, in floating point arithmetic is
                 presented. The algorithm is based on an efficient
                 numerical evaluation of certain closed contour
                 integrals in the complex plane. For large degree $n$,
                 the algorithm is significantly faster than the $ O(n) $
                 complexity of the well known three-term recurrence
                 relation. Comparable accuracy is achieved in no more $
                 O(\sqrt {n}) $ than operations, and for arguments
                 bounded away from $ \pm \sqrt {2n} $, only $ O(\sqrt
                 {\log n}) $ operations.",
  acknowledgement = ack-nhfb,
  fjournal =     "BIT. Numerical Mathematics",
  journal-URL =  "http://link.springer.com/journal/10543",
  keywords =     "fast algorithm; Hermite functions; numerical
                 integration; recursion",
}


bibtype: article
filename: elefunt.bib
label: Chen:2009:SPA
author: Yunfei Chen and Norman C. Beaulieu
title: A simple polynomial approximation to the Gaussian $Q$-function and its application
journal: j-IEEE-COMMUN-LET
volume: 13
number: 2
pages: 124--126
month: feb
monthnumber: 2
year: 2009
CODEN: ICLEF6
DOI: https://doi.org/10.1109/lcomm.2009.081754
ISSN: 1089-7798 (print), 1558-2558 (electronic) OR 10897798 (print), 15582558 (electronic)
ISSNL: 1089-7798 OR 10897798
bibdate: Sat Dec 16 15:46:17 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2017.12.16 15:46:17 ???
URL: http://ieeexplore.ieee.org/document/4783779/
fjournal: IEEE Communications Letters
acknowledgement: ack-nhfb
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=4234
rawauthor: Yunfei Chen and Norman C. Beaulieu
rawtitle: A simple polynomial approximation to the {Gaussian} {$Q$}-function and its application
entry:
@Article{Chen:2009:SPA,
  author =       "Yunfei Chen and Norman C. Beaulieu",
  title =        "A simple polynomial approximation to the {Gaussian}
                 {$Q$}-function and its application",
  journal =      j-IEEE-COMMUN-LET,
  volume =       "13",
  number =       "2",
  pages =        "124--126",
  month =        feb,
  year =         "2009",
  CODEN =        "ICLEF6",
  DOI =          "https://doi.org/10.1109/lcomm.2009.081754",
  ISSN =         "1089-7798 (print), 1558-2558 (electronic)",
  ISSN-L =       "1089-7798",
  bibdate =      "Sat Dec 16 15:46:17 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://ieeexplore.ieee.org/document/4783779/",
  acknowledgement = ack-nhfb,
  fjournal =     "IEEE Communications Letters",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=4234",
}


bibtype: inproceedings
filename: elefunt.bib
label: Chevillard:2009:CFC
author: Sylvain Chevillard and Mioara Joldes and Christoph Lauter
title: Certified and Fast Computation of Supremum Norms of Approximation Errors
crossref: Bruguera:2009:PIS
pages: 169--176
year: 2009
bibdate: Fri Jun 12 12:34:25 2009
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2009.06.12 12:34:25 ???
acknowledgement: ack-nhfb
rawauthor: Sylvain Chevillard and Mioara Joldes and Christoph Lauter
rawtitle: Certified and Fast Computation of Supremum Norms of Approximation Errors
abstract:
In many numerical programs there is a need for a high-quality floating-point approximation of useful functions $f$, such as such as $ exp $, $ sin $, $ erf $. In the actual implementation, the function is replaced by a polynomial $p$, which leads to an approximation error (absolute or relative) $ epsilon = p - f $ or $ epsilon = p / f - 1 $. The tight yet certain bounding of this error is an important step towards safe implementations. The problem is difficult mainly because that approximation error is very small and the difference $ p - f $ is subject to high cancellation. Previous approaches for computing the supremum norm in this degenerate case, have proven to be unsafe, not sufficiently tight or too tedious in manual work. We present a safe and fast algorithm that computes a tight lower and upper bound for the supremum norms of approximation errors. The algorithm is based on a combination of several techniques, including enhanced interval arithmetic, automatic differentiation and isolation of the roots of a polynomial. We have implemented our algorithm and give timings on several examples.

keywords:
approximation error; ARITH-19; automatic/algorithmic differentiation; certified computation; elementary function; interval arithmetic; roots isolation technique.; supremum/infinity norm

rawabstract:
In many numerical programs there is a need for a high-quality floating-point approximation of useful functions $f$, such as such as $ \exp $, $ \sin $, $ \erf $. In the actual implementation, the function is replaced by a polynomial $p$, which leads to an approximation error (absolute or relative) $ \epsilon = p - f $ or $ \epsilon = p / f - 1 $. The tight yet certain bounding of this error is an important step towards safe implementations. The problem is difficult mainly because that approximation error is very small and the difference $ p - f $ is subject to high cancellation. Previous approaches for computing the supremum norm in this degenerate case, have proven to be unsafe, not sufficiently tight or too tedious in manual work. We present a safe and fast algorithm that computes a tight lower and upper bound for the supremum norms of approximation errors. The algorithm is based on a combination of several techniques, including enhanced interval arithmetic, automatic differentiation and isolation of the roots of a polynomial. We have implemented our algorithm and give timings on several examples.

entry:
@InProceedings{Chevillard:2009:CFC,
  author =       "Sylvain Chevillard and Mioara Joldes and Christoph
                 Lauter",
  title =        "Certified and Fast Computation of Supremum Norms of
                 Approximation Errors",
  crossref =     "Bruguera:2009:PIS",
  pages =        "169--176",
  year =         "2009",
  bibdate =      "Fri Jun 12 12:34:25 2009",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  abstract =     "In many numerical programs there is a need for a
                 high-quality floating-point approximation of useful
                 functions $f$, such as such as $ \exp $, $ \sin $, $
                 \erf $. In the actual implementation, the function is
                 replaced by a polynomial $p$, which leads to an
                 approximation error (absolute or relative) $ \epsilon =
                 p - f $ or $ \epsilon = p / f - 1 $. The tight yet
                 certain bounding of this error is an important step
                 towards safe implementations. The problem is difficult
                 mainly because that approximation error is very small
                 and the difference $ p - f $ is subject to high
                 cancellation. Previous approaches for computing the
                 supremum norm in this degenerate case, have proven to
                 be unsafe, not sufficiently tight or too tedious in
                 manual work. We present a safe and fast algorithm that
                 computes a tight lower and upper bound for the supremum
                 norms of approximation errors. The algorithm is based
                 on a combination of several techniques, including
                 enhanced interval arithmetic, automatic differentiation
                 and isolation of the roots of a polynomial. We have
                 implemented our algorithm and give timings on several
                 examples.",
  acknowledgement = ack-nhfb,
  keywords =     "approximation error; ARITH-19; automatic/algorithmic
                 differentiation; certified computation; elementary
                 function; interval arithmetic; roots isolation
                 technique.; supremum/infinity norm",
}


bibtype: techreport
filename: elefunt.bib
label: Chevillard:2009:FEE
author: S. Chevillard
title: The functions ERF and ERFC computed with arbitrary precision
type: Report
number: RRLIP2009-04
institution: HAL
pages: 32
year: 2009
bibdate: Mon Jun 12 16:09:53 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2017.06.12 16:09:53 ???
acknowledgement: ack-nhfb
rawauthor: S. Chevillard
rawtitle: The functions {ERF} and {ERFC} computed with arbitrary precision
entry:
@TechReport{Chevillard:2009:FEE,
  author =       "S. Chevillard",
  title =        "The functions {ERF} and {ERFC} computed with arbitrary
                 precision",
  type =         "Report",
  number =       "RRLIP2009-04",
  institution =  "HAL",
  address =      "????",
  pages =        "32",
  year =         "2009",
  bibdate =      "Mon Jun 12 16:09:53 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
}


bibtype: article
filename: elefunt.bib
label: Daumas:2009:VRN
author: M. Daumas and D. Lester and C. Muoz
title: Verified Real Number Calculations: a Library for Interval Arithmetic
journal: j-IEEE-TRANS-COMPUT
volume: 58
number: 2
pages: 226--237
month: feb
monthnumber: 2
year: 2009
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2008.213
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Fri Jun 12 08:51:00 MDT 2009
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2009.06.12 08:51:00 MDT
fjournal: IEEE Transactions on Computers
keywords: interval arithmetic; proof checking; real number calculations; theorem proving
remark: Extended version of ARITH-18 article cite Daumas:2009:VRN.
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: M. Daumas and D. Lester and C. Muoz
rawtitle: Verified Real Number Calculations: a Library for Interval Arithmetic
abstract:
Real number calculations on elementary functions are remarkably difficult to handle in mechanical proofs. In this paper, we show how these calculations can be performed within a theorem prover or proof assistant in a convenient and highly automated as well as interactive way. First, we formally establish upper and lower bounds for elementary functions. Then, based on these bounds, we develop a rational interval arithmetic where real number calculations take place in an algebraic setting. In order to reduce the dependency effect of interval arithmetic, we integrate two techniques: interval splitting and Taylor series expansions. This pragmatic approach has been developed, and formally verified, in a theorem prover. The formal development also includes a set of customizable strategies to automate proofs involving explicit calculations over real numbers. Our ultimate goal is to provide guaranteed proofs of numerical properties with minimal human theorem-prover interaction.

rawabstract:
Real number calculations on elementary functions are remarkably difficult to handle in mechanical proofs. In this paper, we show how these calculations can be performed within a theorem prover or proof assistant in a convenient and highly automated as well as interactive way. First, we formally establish upper and lower bounds for elementary functions. Then, based on these bounds, we develop a rational interval arithmetic where real number calculations take place in an algebraic setting. In order to reduce the dependency effect of interval arithmetic, we integrate two techniques: interval splitting and Taylor series expansions. This pragmatic approach has been developed, and formally verified, in a theorem prover. The formal development also includes a set of customizable strategies to automate proofs involving explicit calculations over real numbers. Our ultimate goal is to provide guaranteed proofs of numerical properties with minimal human theorem-prover interaction.

entry:
@Article{Daumas:2009:VRN,
  author =       "M. Daumas and D. Lester and C. Muoz",
  title =        "Verified Real Number Calculations: a Library for
                 Interval Arithmetic",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "58",
  number =       "2",
  pages =        "226--237",
  month =        feb,
  year =         "2009",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2008.213",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Fri Jun 12 08:51:00 MDT 2009",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  abstract =     "Real number calculations on elementary functions are
                 remarkably difficult to handle in mechanical proofs. In
                 this paper, we show how these calculations can be
                 performed within a theorem prover or proof assistant in
                 a convenient and highly automated as well as
                 interactive way. First, we formally establish upper and
                 lower bounds for elementary functions. Then, based on
                 these bounds, we develop a rational interval arithmetic
                 where real number calculations take place in an
                 algebraic setting. In order to reduce the dependency
                 effect of interval arithmetic, we integrate two
                 techniques: interval splitting and Taylor series
                 expansions. This pragmatic approach has been developed,
                 and formally verified, in a theorem prover. The formal
                 development also includes a set of customizable
                 strategies to automate proofs involving explicit
                 calculations over real numbers. Our ultimate goal is to
                 provide guaranteed proofs of numerical properties with
                 minimal human theorem-prover interaction.",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
  keywords =     "interval arithmetic; proof checking; real number
                 calculations; theorem proving",
  remark =       "Extended version of ARITH-18 article
                 \cite{Daumas:2009:VRN}.",
}


bibtype: article
filename: elefunt.bib
label: Deano:2009:MAS
author: Alfredo Deano and Nico M. Temme
title: On modified asymptotic series involving confluent hypergeometric functions
journal: j-ELECTRON-TRANS-NUMER-ANAL
volume: 35
pages: 88--103
year: 2009
ISSN: 1068-9613 (print), 1097-4067 (electronic) OR 10689613 (print), 10974067 (electronic)
ISSNL: 1068-9613 OR 10689613
bibdate: Mon Sep 6 12:28:30 MDT 2010
bibsource: http://etna.mcs.kent.edu/; https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2010.09.06 12:28:30 MDT
URL: http://etna.mcs.kent.edu/vol.35.2009/pp88-103.dir/pp88-103.pdf
fjournal: Electronic Transactions on Numerical Analysis
acknowledgement: ack-nhfb
journalURL: http://etna.mcs.kent.edu/
rawauthor: Alfredo Dea{\~n}o and Nico M. Temme
rawtitle: On modified asymptotic series involving confluent hypergeometric functions
entry:
@Article{Deano:2009:MAS,
  author =       "Alfredo Dea{\~n}o and Nico M. Temme",
  title =        "On modified asymptotic series involving confluent
                 hypergeometric functions",
  journal =      j-ELECTRON-TRANS-NUMER-ANAL,
  volume =       "35",
  pages =        "88--103",
  year =         "2009",
  CODEN =        "????",
  ISSN =         "1068-9613 (print), 1097-4067 (electronic)",
  ISSN-L =       "1068-9613",
  bibdate =      "Mon Sep 6 12:28:30 MDT 2010",
  bibsource =    "http://etna.mcs.kent.edu/;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://etna.mcs.kent.edu/vol.35.2009/pp88-103.dir/pp88-103.pdf",
  acknowledgement = ack-nhfb,
  fjournal =     "Electronic Transactions on Numerical Analysis",
  journal-URL =  "http://etna.mcs.kent.edu/",
}


bibtype: article
filename: elefunt.bib
label: FreitasDeAbreu:2009:JCU
author: Giuseppe Thadeu Freitas De Abreu
title: Jensen--Cotes upper and lower bounds on the Gaussian $Q$-function and related functions
journal: j-IEEE-TRANS-COMM
volume: 57
number: 11
pages: 3328--3338
month: nov
monthnumber: 11
year: 2009
CODEN: IECMBT
DOI: https://doi.org/10.1109/tcomm.2009.11.080479
ISSN: 0090-6778 (print), 1558-0857 (electronic) OR 00906778 (print), 15580857 (electronic)
ISSNL: 0090-6778 OR 00906778
bibdate: Sat Dec 16 15:12:46 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2017.12.16 15:12:46 ???
fjournal: IEEE Transactions on Communications
acknowledgement: ack-nhfb
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=26
rawauthor: Giuseppe Thadeu {Freitas De Abreu}
rawtitle: {Jensen--Cotes} upper and lower bounds on the {Gaussian} {$Q$}-function and related functions
entry:
@Article{FreitasDeAbreu:2009:JCU,
  author =       "Giuseppe Thadeu {Freitas De Abreu}",
  title =        "{Jensen--Cotes} upper and lower bounds on the
                 {Gaussian} {$Q$}-function and related functions",
  journal =      j-IEEE-TRANS-COMM,
  volume =       "57",
  number =       "11",
  pages =        "3328--3338",
  month =        nov,
  year =         "2009",
  CODEN =        "IECMBT",
  DOI =          "https://doi.org/10.1109/tcomm.2009.11.080479",
  ISSN =         "0090-6778 (print), 1558-0857 (electronic)",
  ISSN-L =       "0090-6778",
  bibdate =      "Sat Dec 16 15:12:46 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  acknowledgement = ack-nhfb,
  fjournal =     "IEEE Transactions on Communications",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=26",
}


bibtype: article
filename: elefunt.bib
label: Fukushima:2009:FCC
author: Toshio Fukushima
title: Fast computation of complete elliptic integrals and Jacobian elliptic functions
journal: j-CELEST-MECH-DYN-ASTR
volume: 105
number: 4
pages: 305--328
month: dec
monthnumber: 12
year: 2009
CODEN: CLMCAV
DOI: https://doi.org/10.1007/s10569-009-9228-z
ISSN: 0923-2958 (print), 1572-9478 (electronic) OR 09232958 (print), 15729478 (electronic)
ISSNL: 0923-2958 OR 09232958
MRclass: 33E05 (33F05 65E05)
MRnumber: 2559416
MRreviewer: Mehdi Hassani
bibdate: Wed Oct 20 21:29:31 2010
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2010.10.20 21:29:31 ???
URL: http://www.springerlink.com/content/0923-2958/
fjournal: Celestial Mechanics & Dynamical Astronomy. An International Journal of Space Dynamics
acknowledgement: ack-nhfb
rawauthor: Toshio Fukushima
rawtitle: Fast computation of complete elliptic integrals and {Jacobian} elliptic functions
abstract:
As a preparation step to compute Jacobian elliptic functions efficiently, we created a fast method to calculate the complete elliptic integral of the first and second kinds, $ K(m) $ and $ E(m) $, for the standard domain of the elliptic parameter, $ 0 < m < 1 $. For the case $ 0 < m < 0.9 $, the method utilizes $ 10 $ pairs of approximate polynomials of the order of $9$--$ 19 $ obtained by truncating Taylor series expansions of the integrals. Otherwise, the associate integrals, $ K(1 - m) $ and $ E(1 - m) $, are first computed by a pair of the approximate polynomials and then transformed to $ K(m) $ and $ E(m) $ by means of Jacobi's nome, $q$, and Legendre's identity relation. In average, the new method runs more-than-twice faster than the existing methods including Cody's Chebyshev polynomial approximation of Hastings type and Innes' formulation based on $q$-series expansions. Next, we invented a fast procedure to compute simultaneously three Jacobian elliptic functions, tt sn(u|m), tt cn(u|m), and tt dn(u|m), by repeated usage of the double argument formulae starting from the Maclaurin series expansions with respect to the elliptic argument, $u$, after its domain is reduced to the standard range, $ 0 leq u < K(m) / 4 $, with the help of the new method to compute K(m). The new procedure is 25--70% faster than the methods based on the Gauss transformation such as Bulirsch's algorithm, sncndn, quoted in the Numerical Recipes even if the acceleration of computation of $ K(m) $ is not taken into account.

keywords:
complete elliptic integrals; Encke's method; Innes' method; Jacobian elliptic functions; nome expansion; numerical methods

rawabstract:
As a preparation step to compute Jacobian elliptic functions efficiently, we created a fast method to calculate the complete elliptic integral of the first and second kinds, $ K(m) $ and $ E(m) $, for the standard domain of the elliptic parameter, $ 0 < m < 1 $. For the case $ 0 < m < 0.9 $, the method utilizes $ 10 $ pairs of approximate polynomials of the order of $9$--$ 19 $ obtained by truncating Taylor series expansions of the integrals. Otherwise, the associate integrals, $ K(1 - m) $ and $ E(1 - m) $, are first computed by a pair of the approximate polynomials and then transformed to $ K(m) $ and $ E(m) $ by means of Jacobi's nome, $q$, and Legendre's identity relation. In average, the new method runs more-than-twice faster than the existing methods including Cody's Chebyshev polynomial approximation of Hastings type and Innes' formulation based on $q$-series expansions. Next, we invented a fast procedure to compute simultaneously three Jacobian elliptic functions, {\tt sn(u|m)}, {\tt cn(u|m)}, and {\tt dn(u|m)}, by repeated usage of the double argument formulae starting from the Maclaurin series expansions with respect to the elliptic argument, $u$, after its domain is reduced to the standard range, $ 0 \leq u < K(m) / 4 $, with the help of the new method to compute K(m). The new procedure is 25--70\% faster than the methods based on the Gauss transformation such as Bulirsch's algorithm, sncndn, quoted in the Numerical Recipes even if the acceleration of computation of $ K(m) $ is not taken into account.

entry:
@Article{Fukushima:2009:FCC,
  author =       "Toshio Fukushima",
  title =        "Fast computation of complete elliptic integrals and
                 {Jacobian} elliptic functions",
  journal =      j-CELEST-MECH-DYN-ASTR,
  volume =       "105",
  number =       "4",
  pages =        "305--328",
  month =        dec,
  year =         "2009",
  CODEN =        "CLMCAV",
  DOI =          "https://doi.org/10.1007/s10569-009-9228-z",
  ISSN =         "0923-2958 (print), 1572-9478 (electronic)",
  ISSN-L =       "0923-2958",
  MRclass =      "33E05 (33F05 65E05)",
  MRnumber =     "2559416",
  MRreviewer =   "Mehdi Hassani",
  bibdate =      "Wed Oct 20 21:29:31 2010",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.springerlink.com/content/0923-2958/",
  abstract =     "As a preparation step to compute Jacobian elliptic
                 functions efficiently, we created a fast method to
                 calculate the complete elliptic integral of the first
                 and second kinds, $ K(m) $ and $ E(m) $, for the
                 standard domain of the elliptic parameter, $ 0 < m < 1
                 $. For the case $ 0 < m < 0.9 $, the method utilizes $
                 10 $ pairs of approximate polynomials of the order of
                 $9$--$ 19 $ obtained by truncating Taylor series
                 expansions of the integrals. Otherwise, the associate
                 integrals, $ K(1 - m) $ and $ E(1 - m) $, are first
                 computed by a pair of the approximate polynomials and
                 then transformed to $ K(m) $ and $ E(m) $ by means of
                 Jacobi's nome, $q$, and Legendre's identity relation.
                 In average, the new method runs more-than-twice faster
                 than the existing methods including Cody's Chebyshev
                 polynomial approximation of Hastings type and Innes'
                 formulation based on $q$-series expansions. Next, we
                 invented a fast procedure to compute simultaneously
                 three Jacobian elliptic functions, {\tt sn(u|m)}, {\tt
                 cn(u|m)}, and {\tt dn(u|m)}, by repeated usage of the
                 double argument formulae starting from the Maclaurin
                 series expansions with respect to the elliptic
                 argument, $u$, after its domain is reduced to the
                 standard range, $ 0 \leq u < K(m) / 4 $, with the help
                 of the new method to compute K(m). The new procedure is
                 25--70\% faster than the methods based on the Gauss
                 transformation such as Bulirsch's algorithm, sncndn,
                 quoted in the Numerical Recipes even if the
                 acceleration of computation of $ K(m) $ is not taken
                 into account.",
  acknowledgement = ack-nhfb,
  fjournal =     "Celestial Mechanics \& Dynamical Astronomy. An
                 International Journal of Space Dynamics",
  keywords =     "complete elliptic integrals; Encke's method; Innes'
                 method; Jacobian elliptic functions; nome expansion;
                 numerical methods",
}


bibtype: article
filename: elefunt.bib
label: Fukushima:2009:FCJ
author: Toshio Fukushima
journal: j-CELEST-MECH-DYN-ASTR
volume: 105
number: 1--3
pages: 245--260
year: 2009
CODEN: CLMCAV
DOI: https://doi.org/10.1007/s10569-008-9177-y
ISSN: 0923-2958 (print), 1572-9478 (electronic) OR 09232958 (print), 15729478 (electronic)
ISSNL: 0923-2958 OR 09232958
MRclass: 33E05 (33F05 65D20 70M20)
MRnumber: 2551836
bibdate: Mon Oct 24 11:37:20 2011
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2011.10.24 11:37:20 ???
URL: http://www.springerlink.com/content/0923-2958/
fjournal: Celestial Mechanics & Dynamical Astronomy. An International Journal of Space Dynamics
keywords: elliptic integrals; extended body dynamics; Jacobian elliptic functions; numerical method; rotation
acknowledgement: ack-nhfb
rawauthor: Toshio Fukushima
title:
Fast computation of Jacobian elliptic functions and incomplete elliptic integrals for constant values of elliptic parameter and elliptic characteristic

abstract:
In order to accelerate the numerical evaluation of torque-free rotation of triaxial rigid bodies, we present a fast method to compute various kinds of elliptic functions for a series of the elliptic argument when the elliptic parameter and the elliptic characteristic are fixed. The functions we evaluate are the Jacobian elliptic functions and the incomplete elliptic integral of the second and third kinds regarded as a function of that of the first kind. The key technique is the utilization of the Maclaurin series expansion and the addition theorems with respect to the elliptic argument. The new method is around 25 times faster than the method using the incomplete elliptic integral of general kind and around 70 times faster than the method using mathematical libraries given in the latest version of Numerical Recipes.

rawabstract:
In order to accelerate the numerical evaluation of torque-free rotation of triaxial rigid bodies, we present a fast method to compute various kinds of elliptic functions for a series of the elliptic argument when the elliptic parameter and the elliptic characteristic are fixed. The functions we evaluate are the Jacobian elliptic functions and the incomplete elliptic integral of the second and third kinds regarded as a function of that of the first kind. The key technique is the utilization of the Maclaurin series expansion and the addition theorems with respect to the elliptic argument. The new method is around 25 times faster than the method using the incomplete elliptic integral of general kind and around 70 times faster than the method using mathematical libraries given in the latest version of Numerical Recipes.

rawtitle:
Fast computation of {Jacobian} elliptic functions and incomplete elliptic integrals for constant values of elliptic parameter and elliptic characteristic

entry:
@Article{Fukushima:2009:FCJ,
  author =       "Toshio Fukushima",
  title =        "Fast computation of {Jacobian} elliptic functions and
                 incomplete elliptic integrals for constant values of
                 elliptic parameter and elliptic characteristic",
  journal =      j-CELEST-MECH-DYN-ASTR,
  volume =       "105",
  number =       "1--3",
  pages =        "245--260",
  year =         "2009",
  CODEN =        "CLMCAV",
  DOI =          "https://doi.org/10.1007/s10569-008-9177-y",
  ISSN =         "0923-2958 (print), 1572-9478 (electronic)",
  ISSN-L =       "0923-2958",
  MRclass =      "33E05 (33F05 65D20 70M20)",
  MRnumber =     "2551836",
  bibdate =      "Mon Oct 24 11:37:20 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://www.springerlink.com/content/0923-2958/",
  abstract =     "In order to accelerate the numerical evaluation of
                 torque-free rotation of triaxial rigid bodies, we
                 present a fast method to compute various kinds of
                 elliptic functions for a series of the elliptic
                 argument when the elliptic parameter and the elliptic
                 characteristic are fixed. The functions we evaluate are
                 the Jacobian elliptic functions and the incomplete
                 elliptic integral of the second and third kinds
                 regarded as a function of that of the first kind. The
                 key technique is the utilization of the Maclaurin
                 series expansion and the addition theorems with respect
                 to the elliptic argument. The new method is around 25
                 times faster than the method using the incomplete
                 elliptic integral of general kind and around 70 times
                 faster than the method using mathematical libraries
                 given in the latest version of Numerical Recipes.",
  acknowledgement = ack-nhfb,
  fjournal =     "Celestial Mechanics \& Dynamical Astronomy. An
                 International Journal of Space Dynamics",
  keywords =     "elliptic integrals; extended body dynamics; Jacobian
                 elliptic functions; numerical method; rotation",
}


bibtype: article
filename: elefunt.bib
label: Guo:2009:CLC
author: Senlin Guo and Feng Qi
title: A class of logarithmically completely monotonic functions associated with the gamma function
journal: j-J-COMPUT-APPL-MATH
volume: 224
number: 1
pages: 127--132
day: 1
month: feb
monthnumber: 2
year: 2009
CODEN: JCAMDI
ISSN: 0377-0427 (print), 1879-1778 (electronic) OR 03770427 (print), 18791778 (electronic)
ISSNL: 0377-0427 OR 03770427
bibdate: Sat Feb 25 13:13:29 MST 2017
bibtimestamp: 2017.02.25 13:13:29 MST
URL: http://www.sciencedirect.com/science/article/pii/S0377042708001829
fjournal: Journal of Computational and Applied Mathematics
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03770427
rawauthor: Senlin Guo and Feng Qi
rawtitle: A class of logarithmically completely monotonic functions associated with the gamma function
entry:
@Article{Guo:2009:CLC,
  author =       "Senlin Guo and Feng Qi",
  title =        "A class of logarithmically completely monotonic
                 functions associated with the gamma function",
  journal =      j-J-COMPUT-APPL-MATH,
  volume =       "224",
  number =       "1",
  pages =        "127--132",
  day =          "1",
  month =        feb,
  year =         "2009",
  CODEN =        "JCAMDI",
  ISSN =         "0377-0427 (print), 1879-1778 (electronic)",
  ISSN-L =       "0377-0427",
  bibdate =      "Sat Feb 25 13:13:29 MST 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/jcomputapplmath2005.bib",
  URL =          "http://www.sciencedirect.com/science/article/pii/S0377042708001829",
  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: Guralnik:2009:ISV
author: Elena Guralnik and Ariel J. Birnbaum and Anatoly Koyfman and Avi Kaplan
title: Implementation Specific Verification of Divide and Square Root Instructions
crossref: Bruguera:2009:PIS
pages: 114--121
year: 2009
bibdate: Fri Jun 12 12:34:25 2009
bibtimestamp: 2009.06.12 12:34:25 ???
keywords: ARITH-19
acknowledgement: ack-nhfb
rawauthor: Elena Guralnik and Ariel J. Birnbaum and Anatoly Koyfman and Avi Kaplan
rawtitle: Implementation Specific Verification of Divide and Square Root Instructions
abstract:
Floating point operations such as divide and square root are typically implemented in microcode rather than dedicated logic. Bugs in these operations missed by generic black-box verification tools, were analyzed. This led to the conclusion that the corner cases, in addition to being implementation dependent, could not be characterized in terms of special input or output values in a straightforward manner.par However, many of those cases can be easily generalized for many known implementations. The typical implementation uses a known iterative approximation algorithm, such as the Newton--Raphson method, to calculate the desired result; thus, it is sufficient to produce the corner cases associated with the specific algorithm.par We investigated the following problem: given an iterative algorithm to compute a binary floating point operation, the iteration number, and an interval, find random inputs for the operation that, after the requested iteration, yield a relative error within the specified interval. This paper describes a method to solve this problem. This method was implemented in a floating-point test generator and is currently being used to verify the floating-point units of several processors.

rawabstract:
Floating point operations such as divide and square root are typically implemented in microcode rather than dedicated logic. Bugs in these operations missed by generic black-box verification tools, were analyzed. This led to the conclusion that the corner cases, in addition to being implementation dependent, could not be characterized in terms of special input or output values in a straightforward manner.\par However, many of those cases can be easily generalized for many known implementations. The typical implementation uses a known iterative approximation algorithm, such as the Newton--Raphson method, to calculate the desired result; thus, it is sufficient to produce the corner cases associated with the specific algorithm.\par We investigated the following problem: given an iterative algorithm to compute a binary floating point operation, the iteration number, and an interval, find random inputs for the operation that, after the requested iteration, yield a relative error within the specified interval. This paper describes a method to solve this problem. This method was implemented in a floating-point test generator and is currently being used to verify the floating-point units of several processors.

entry:
@InProceedings{Guralnik:2009:ISV,
  author =       "Elena Guralnik and Ariel J. Birnbaum and Anatoly
                 Koyfman and Avi Kaplan",
  title =        "Implementation Specific Verification of Divide and
                 Square Root Instructions",
  crossref =     "Bruguera:2009:PIS",
  pages =        "114--121",
  year =         "2009",
  bibdate =      "Fri Jun 12 12:34:25 2009",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/fparith.bib",
  abstract =     "Floating point operations such as divide and square
                 root are typically implemented in microcode rather than
                 dedicated logic. Bugs in these operations missed by
                 generic black-box verification tools, were analyzed.
                 This led to the conclusion that the corner cases, in
                 addition to being implementation dependent, could not
                 be characterized in terms of special input or output
                 values in a straightforward manner.\par

                 However, many of those cases can be easily generalized
                 for many known implementations. The typical
                 implementation uses a known iterative approximation
                 algorithm, such as the Newton--Raphson method, to
                 calculate the desired result; thus, it is sufficient to
                 produce the corner cases associated with the specific
                 algorithm.\par

                 We investigated the following problem: given an
                 iterative algorithm to compute a binary floating point
                 operation, the iteration number, and an interval, find
                 random inputs for the operation that, after the
                 requested iteration, yield a relative error within the
                 specified interval. This paper describes a method to
                 solve this problem. This method was implemented in a
                 floating-point test generator and is currently being
                 used to verify the floating-point units of several
                 processors.",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH-19",
}


bibtype: article
filename: elefunt.bib
label: Han:2009:ICS
author: Dong-Guk Han and Dooho Choi and Howon Kim
title: Improved Computation of Square Roots in Specific Finite Fields
journal: j-IEEE-TRANS-COMPUT
volume: 58
number: 2
pages: 188--196
month: feb
monthnumber: 2
year: 2009
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2008.201
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Mon Jul 4 11:37:39 MDT 2011
bibtimestamp: 2011.07.04 11:37:39 MDT
URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4663058
fjournal: IEEE Transactions on Computers
keywords: cryptography; efficient computation; finite fields; square roots
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: Dong-Guk Han and Dooho Choi and Howon Kim
rawtitle: Improved Computation of Square Roots in Specific Finite Fields
bibsource:
https://www.math.utah.edu/pub/tex/bib/cryptography2000.bib; 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/ieeetranscomput2000.bib

abstract:
In this paper, we study exponentiation in the specific finite fields $ F_q $ with very special exponents such as those that occur in algorithms for computing square roots. Here, $q$ is a prime power, $ q = p^k $, where $ k > 1 $, and $k$ is odd. Our algorithmic approach improves the corresponding exponentiation resulted from the better rewritten exponent. To the best of our knowledge, it is the first major improvement to the Tonelli--Shanks algorithm, for example, the number of multiplications can be reduced to at least 60 percent on the average when $ p equiv 1 pmod 16 $. Several numerical examples are given that show the speedup of the proposed methods.

rawabstract:
In this paper, we study exponentiation in the specific finite fields $ F_q $ with very special exponents such as those that occur in algorithms for computing square roots. Here, $q$ is a prime power, $ q = p^k $, where $ k > 1 $, and $k$ is odd. Our algorithmic approach improves the corresponding exponentiation resulted from the better rewritten exponent. To the best of our knowledge, it is the first major improvement to the Tonelli--Shanks algorithm, for example, the number of multiplications can be reduced to at least 60 percent on the average when $ p \equiv 1 \pmod 16 $. Several numerical examples are given that show the speedup of the proposed methods.

entry:
@Article{Han:2009:ICS,
  author =       "Dong-Guk Han and Dooho Choi and Howon Kim",
  title =        "Improved Computation of Square Roots in Specific
                 Finite Fields",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "58",
  number =       "2",
  pages =        "188--196",
  month =        feb,
  year =         "2009",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2008.201",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Mon Jul 4 11:37:39 MDT 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/cryptography2000.bib;
                 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/ieeetranscomput2000.bib",
  URL =          "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4663058",
  abstract =     "In this paper, we study exponentiation in the specific
                 finite fields $ F_q $ with very special exponents such
                 as those that occur in algorithms for computing square
                 roots. Here, $q$ is a prime power, $ q = p^k $, where $
                 k > 1 $, and $k$ is odd. Our algorithmic approach
                 improves the corresponding exponentiation resulted from
                 the better rewritten exponent. To the best of our
                 knowledge, it is the first major improvement to the
                 Tonelli--Shanks algorithm, for example, the number of
                 multiplications can be reduced to at least 60 percent
                 on the average when $ p \equiv 1 \pmod 16 $. Several
                 numerical examples are given that show the speedup of
                 the proposed methods.",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
  keywords =     "cryptography; efficient computation; finite fields;
                 square roots",
}


bibtype: article
filename: elefunt.bib
label: Harris:2009:MIB
author: Frank E. Harris and J. G. Fripiat
title: Methods for incomplete Bessel function evaluation
journal: j-IJQC
volume: 109
number: 8
pages: 1728--1740
month: feb
monthnumber: 2
year: 2009
CODEN: IJQCB2
DOI: https://doi.org/10.1002/qua.21972
ISSN: 0020-7608 (print), 1097-461X (electronic) OR 00207608 (print), 1097461X (electronic)
ISSNL: 0020-7608 OR 00207608
bibdate: Fri Mar 27 07:41:18 2009
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2009.03.27 07:41:18 ???
fjournal: International Journal of Quantum Chemistry
acknowledgement: ack-nhfb
journalURL: http://www.interscience.wiley.com/jpages/0020-7608/
rawauthor: Frank E. Harris and J. G. Fripiat
rawtitle: Methods for incomplete {Bessel} function evaluation
abstract:
Presented here are detailed methods for evaluating the incomplete Bessel functions arising when Gaussian-type orbitals are used for systems periodic in one spatial dimension. The scheme is designed to yield these incomplete Bessel functions with an absolute accuracy of $ pm 1 times 10^-10 $, for the range of integer orders $ 0 leq n leq 12 $ [a range sufficient for a basis whose members have angular momenta of up to three units ($s$, $p$, $d$, or $f$ atomic functions)]. To reach this accuracy level within acceptable computation times, new rational approximations were developed to compute the special functions involved, namely, the exponential integral $ E_1 (x) $ and the modified Bessel functions $ K_0 (x) $ and $ K_1 (x) $, to absolute accuracy $ pm 1 times 10^-15 $.

keywords:
E1(x); exponential integral; incomplete Bessel function; K0(x); K1(x); leaky aquifer function; modified Bessel function; numerical methods

rawabstract:
Presented here are detailed methods for evaluating the incomplete Bessel functions arising when Gaussian-type orbitals are used for systems periodic in one spatial dimension. The scheme is designed to yield these incomplete Bessel functions with an absolute accuracy of $ \pm 1 \times 10^{-10} $, for the range of integer orders $ 0 \leq n \leq 12 $ [a range sufficient for a basis whose members have angular momenta of up to three units ($s$, $p$, $d$, or $f$ atomic functions)]. To reach this accuracy level within acceptable computation times, new rational approximations were developed to compute the special functions involved, namely, the exponential integral $ E_1 (x) $ and the modified Bessel functions $ K_0 (x) $ and $ K_1 (x) $, to absolute accuracy $ \pm 1 \times 10^{-15} $.

entry:
@Article{Harris:2009:MIB,
  author =       "Frank E. Harris and J. G. Fripiat",
  title =        "Methods for incomplete {Bessel} function evaluation",
  journal =      j-IJQC,
  volume =       "109",
  number =       "8",
  pages =        "1728--1740",
  month =        feb,
  year =         "2009",
  CODEN =        "IJQCB2",
  DOI =          "https://doi.org/10.1002/qua.21972",
  ISSN =         "0020-7608 (print), 1097-461X (electronic)",
  ISSN-L =       "0020-7608",
  bibdate =      "Fri Mar 27 07:41:18 2009",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  abstract =     "Presented here are detailed methods for evaluating the
                 incomplete Bessel functions arising when Gaussian-type
                 orbitals are used for systems periodic in one spatial
                 dimension. The scheme is designed to yield these
                 incomplete Bessel functions with an absolute accuracy
                 of $ \pm 1 \times 10^{-10} $, for the range of integer
                 orders $ 0 \leq n \leq 12 $ [a range sufficient for a
                 basis whose members have angular momenta of up to three
                 units ($s$, $p$, $d$, or $f$ atomic functions)]. To
                 reach this accuracy level within acceptable computation
                 times, new rational approximations were developed to
                 compute the special functions involved, namely, the
                 exponential integral $ E_1 (x) $ and the modified
                 Bessel functions $ K_0 (x) $ and $ K_1 (x) $, to
                 absolute accuracy $ \pm 1 \times 10^{-15} $.",
  acknowledgement = ack-nhfb,
  fjournal =     "International Journal of Quantum Chemistry",
  journal-URL =  "http://www.interscience.wiley.com/jpages/0020-7608/",
  keywords =     "E1(x); exponential integral; incomplete Bessel
                 function; K0(x); K1(x); leaky aquifer function;
                 modified Bessel function; numerical methods",
}


bibtype: inproceedings
filename: elefunt.bib
label: Harrison:2009:DTB
author: John Harrison
title: Decimal Transcendentals via Binary
crossref: Bruguera:2009:PIS
pages: 187--194
year: 2009
DOI: https://doi.org/10.1109/ARITH.2009.32
bibdate: Fri Jun 12 12:34:25 2009
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2009.06.12 12:34:25 ???
keywords: ARITH-19
acknowledgement: ack-nhfb
rawauthor: John Harrison
rawtitle: Decimal Transcendentals via Binary
abstract:
We describe the design and implementation of a comprehensive library of transcendental functions for the new IEEE decimal floating-point formats. In principle, such functions are very much analogous to their binary counterparts, though with a few additional subtleties connected with `scale' (preferred exponent). But our approach has been not to employ direct techniques, but rather to re-use existing binary functions as much as possible, both for greater efficiency and ease of implementation. For some functions the most straightforward approach (convert from decimal to binary, perform binary operation, convert back) works well. In many cases, however, these are insufficiently accurate, and subtler approaches must be used.

rawabstract:
We describe the design and implementation of a comprehensive library of transcendental functions for the new IEEE decimal floating-point formats. In principle, such functions are very much analogous to their binary counterparts, though with a few additional subtleties connected with `scale' (preferred exponent). But our approach has been not to employ direct techniques, but rather to re-use existing binary functions as much as possible, both for greater efficiency and ease of implementation. For some functions the most straightforward approach (convert from decimal to binary, perform binary operation, convert back) works well. In many cases, however, these are insufficiently accurate, and subtler approaches must be used.

entry:
@InProceedings{Harrison:2009:DTB,
  author =       "John Harrison",
  title =        "Decimal Transcendentals via Binary",
  crossref =     "Bruguera:2009:PIS",
  pages =        "187--194",
  year =         "2009",
  DOI =          "https://doi.org/10.1109/ARITH.2009.32",
  bibdate =      "Fri Jun 12 12:34:25 2009",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  abstract =     "We describe the design and implementation of a
                 comprehensive library of transcendental functions for
                 the new IEEE decimal floating-point formats. In
                 principle, such functions are very much analogous to
                 their binary counterparts, though with a few additional
                 subtleties connected with `scale' (preferred exponent).
                 But our approach has been not to employ direct
                 techniques, but rather to re-use existing binary
                 functions as much as possible, both for greater
                 efficiency and ease of implementation. For some
                 functions the most straightforward approach (convert
                 from decimal to binary, perform binary operation,
                 convert back) works well. In many cases, however, these
                 are insufficiently accurate, and subtler approaches
                 must be used.",
  acknowledgement = ack-nhfb,
  keywords =     "ARITH-19",
}


bibtype: inproceedings
filename: elefunt.bib
label: Harrison:2009:FAB
author: John Harrison
title: Fast and Accurate Bessel Function Computation
crossref: Bruguera:2009:PIS
pages: 104--113
year: 2009
bibdate: Fri Jun 12 12:34:25 2009
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2009.06.12 12:34:25 ???
keywords: $J0(x), J1(1), Y0(x), Y1(1)$; ARITH-19; ordinary Bessel functions of the first and second kinds
acknowledgement: ack-nhfb
rawauthor: John Harrison
rawtitle: Fast and Accurate {Bessel} Function Computation
abstract:
The Bessel functions are considered relatively difficult to compute. Although they have a simple power series expansion that is everywhere convergent, they exhibit approximately periodic behavior which makes the direct use of the power series impractically slow and numerically unstable. We describe an alternative method based on systematic expansion around the zeros, refining existing techniques based on Hankel expansions, which mostly avoids the use of multiprecision arithmetic while yielding accurate results.

rawabstract:
The Bessel functions are considered relatively difficult to compute. Although they have a simple power series expansion that is everywhere convergent, they exhibit approximately periodic behavior which makes the direct use of the power series impractically slow and numerically unstable. We describe an alternative method based on systematic expansion around the zeros, refining existing techniques based on Hankel expansions, which mostly avoids the use of multiprecision arithmetic while yielding accurate results.

entry:
@InProceedings{Harrison:2009:FAB,
  author =       "John Harrison",
  title =        "Fast and Accurate {Bessel} Function Computation",
  crossref =     "Bruguera:2009:PIS",
  pages =        "104--113",
  year =         "2009",
  bibdate =      "Fri Jun 12 12:34:25 2009",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  abstract =     "The Bessel functions are considered relatively
                 difficult to compute. Although they have a simple power
                 series expansion that is everywhere convergent, they
                 exhibit approximately periodic behavior which makes the
                 direct use of the power series impractically slow and
                 numerically unstable. We describe an alternative method
                 based on systematic expansion around the zeros,
                 refining existing techniques based on Hankel
                 expansions, which mostly avoids the use of
                 multiprecision arithmetic while yielding accurate
                 results.",
  acknowledgement = ack-nhfb,
  keywords =     "$J0(x), J1(1), Y0(x), Y1(1)$; ARITH-19; ordinary
                 Bessel functions of the first and second kinds",
}


bibtype: book
filename: elefunt.bib
label: Henner:2009:MMP
author: Victor Henner and Tatyana Belozerova and Kyle Forinash
publisher: pub-A-K-PETERS
address: pub-A-K-PETERS:adr
pages: xviii + 841
year: 2009
DOI: https://doi.org/10.1201/b10695
ISBN: 1-56881-335-X OR 156881335X
ISBN13: 978-1-56881-335-6 OR 9781568813356
LCCN: QC20 .H487 2009
bibdate: Sat Oct 30 17:39:29 MDT 2010
bibtimestamp: 2010.10.30 17:39:29 MDT
subject: Mathematical physics; Textbooks
acknowledgement: ack-nhfb
rawauthor: Victor Henner and Tatyana Belozerova and Kyle Forinash
title:
Mathematical Methods in Physics: Partial Differential Equations, Fourier Series, and Special Functions

TOC:
1: Fourier Series 2: Sturm--Liouville Theory 3: One-Dimensional Hyperbolic Equations 4: Two-Dimensional Hyperbolic Equations 5: One-Dimensional Parabolic Equations 6: Parabolic Equations for Higher-Dimensional Problems 7: Elliptic Equations 8: Bessel Functions 9: Legendre Functions A: Eigenvalues and Eigen functions of the Sturm--Liouville Problem B: Auxiliary Functions for Different Types of Boundary Conditions C: The Sturm--Liouville Problem and the Laplace Equation D: Vector Calculus E: How to Use the Software Associated with this Book

rawtitle:
Mathematical Methods in Physics: Partial Differential Equations, {Fourier} Series, and Special Functions

rawTOC:
1: Fourier Series \\ 2: Sturm--Liouville Theory \\ 3: One-Dimensional Hyperbolic Equations \\ 4: Two-Dimensional Hyperbolic Equations \\ 5: One-Dimensional Parabolic Equations \\ 6: Parabolic Equations for Higher-Dimensional Problems \\ 7: Elliptic Equations \\ 8: Bessel Functions \\ 9: Legendre Functions \\ A: Eigenvalues and Eigen functions of the Sturm--Liouville Problem \\ B: Auxiliary Functions for Different Types of Boundary Conditions \\ C: The Sturm--Liouville Problem and the Laplace Equation \\ D: Vector Calculus \\ E: How to Use the Software Associated with this Book

entry:
@Book{Henner:2009:MMP,
  author =       "Victor Henner and Tatyana Belozerova and Kyle
                 Forinash",
  title =        "Mathematical Methods in Physics: Partial Differential
                 Equations, {Fourier} Series, and Special Functions",
  publisher =    pub-A-K-PETERS,
  address =      pub-A-K-PETERS:adr,
  pages =        "xviii + 841",
  year =         "2009",
  DOI =          "https://doi.org/10.1201/b10695",
  ISBN =         "1-56881-335-X",
  ISBN-13 =      "978-1-56881-335-6",
  LCCN =         "QC20 .H487 2009",
  bibdate =      "Sat Oct 30 17:39:29 MDT 2010",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/mathcw.bib;
                 z3950.loc.gov:7090/Voyager",
  acknowledgement = ack-nhfb,
  subject =      "Mathematical physics; Textbooks",
  tableofcontents = "1: Fourier Series \\
                 2: Sturm--Liouville Theory \\
                 3: One-Dimensional Hyperbolic Equations \\
                 4: Two-Dimensional Hyperbolic Equations \\
                 5: One-Dimensional Parabolic Equations \\
                 6: Parabolic Equations for Higher-Dimensional Problems
                 \\
                 7: Elliptic Equations \\
                 8: Bessel Functions \\
                 9: Legendre Functions \\
                 A: Eigenvalues and Eigen functions of the
                 Sturm--Liouville Problem \\
                 B: Auxiliary Functions for Different Types of Boundary
                 Conditions \\
                 C: The Sturm--Liouville Problem and the Laplace
                 Equation \\
                 D: Vector Calculus \\
                 E: How to Use the Software Associated with this Book",
}


bibtype: article
filename: elefunt.bib
label: Lauter:2009:ERB
author: C. Q. Lauter and V. Lefevre
title: An Efficient Rounding Boundary Test for tt pow(x, y) in Double Precision
journal: j-IEEE-TRANS-COMPUT
volume: 58
number: 2
pages: 197--207
month: feb
monthnumber: 2
year: 2009
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2008.202
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Fri Jun 12 08:51:00 MDT 2009
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2009.06.12 08:51:00 MDT
fjournal: IEEE Transactions on Computers
keywords: correct rounding; floating-point arithmetic; power function.
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: C. Q. Lauter and V. Lefevre
rawtitle: An Efficient Rounding Boundary Test for {\tt pow(x, y)} in Double Precision
abstract:
The correct rounding of the function $ textrm pow : (x, y) rightarrow x^y $ is currently based on Ziv's iterative approximation process. In order to ensure its termination, cases when $ x^y $ falls on a rounding-boundary must be filtered out. Such rounding-boundaries are floating-point numbers and midpoints between two consecutive floating-point numbers. Detecting rounding-boundaries for pow is a difficult problem. Previous approaches use repeated square root extraction followed by repeated square and multiply. This paper presents a new rounding-boundary test for pow in double precision, which reduces this to a few comparisons with precomputed constants. These constants are deduced from worst cases for the Table Maker's Dilemma, searched over a small subset of the input domain. This is a novel use of such worst-case bounds. The resulting algorithm has been designed for a fast-on-average correctly rounded implementation of pow, considering the scarcity of rounding-boundary cases. It does not stall average computations for rounding-boundary detection. This paper includes its correctness proof and experimental results.

rawabstract:
The correct rounding of the function $ \textrm {pow} : (x, y) \rightarrow x^y $ is currently based on Ziv's iterative approximation process. In order to ensure its termination, cases when $ x^y $ falls on a rounding-boundary must be filtered out. Such rounding-boundaries are floating-point numbers and midpoints between two consecutive floating-point numbers. Detecting rounding-boundaries for pow is a difficult problem. Previous approaches use repeated square root extraction followed by repeated square and multiply. This paper presents a new rounding-boundary test for pow in double precision, which reduces this to a few comparisons with precomputed constants. These constants are deduced from worst cases for the Table Maker's Dilemma, searched over a small subset of the input domain. This is a novel use of such worst-case bounds. The resulting algorithm has been designed for a fast-on-average correctly rounded implementation of pow, considering the scarcity of rounding-boundary cases. It does not stall average computations for rounding-boundary detection. This paper includes its correctness proof and experimental results.

entry:
@Article{Lauter:2009:ERB,
  author =       "C. Q. Lauter and V. Lefevre",
  title =        "An Efficient Rounding Boundary Test for {\tt pow(x,
                 y)} in Double Precision",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "58",
  number =       "2",
  pages =        "197--207",
  month =        feb,
  year =         "2009",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2008.202",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Fri Jun 12 08:51:00 MDT 2009",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  abstract =     "The correct rounding of the function $ \textrm {pow} :
                 (x, y) \rightarrow x^y $ is currently based on Ziv's
                 iterative approximation process. In order to ensure its
                 termination, cases when $ x^y $ falls on a
                 rounding-boundary must be filtered out. Such
                 rounding-boundaries are floating-point numbers and
                 midpoints between two consecutive floating-point
                 numbers. Detecting rounding-boundaries for pow is a
                 difficult problem. Previous approaches use repeated
                 square root extraction followed by repeated square and
                 multiply. This paper presents a new rounding-boundary
                 test for pow in double precision, which reduces this to
                 a few comparisons with precomputed constants. These
                 constants are deduced from worst cases for the Table
                 Maker's Dilemma, searched over a small subset of the
                 input domain. This is a novel use of such worst-case
                 bounds. The resulting algorithm has been designed for a
                 fast-on-average correctly rounded implementation of
                 pow, considering the scarcity of rounding-boundary
                 cases. It does not stall average computations for
                 rounding-boundary detection. This paper includes its
                 correctness proof and experimental results.",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
  keywords =     "correct rounding; floating-point arithmetic; power
                 function.",
}


bibtype: article
filename: elefunt.bib
label: Linhart:2009:ACL
author: Jean Marie Linhart
title: Algorithm 885: Computing the Logarithm of the Normal Distribution
journal: j-TOMS
volume: 35
number: 3
pages: 20:1--20:10
month: oct
monthnumber: 10
year: 2009
CODEN: ACMSCU
DOI: https://doi.org/10.1145/1391989.1391993
ISSN: 0098-3500 (print), 1557-7295 (electronic) OR 00983500 (print), 15577295 (electronic)
ISSNL: 0098-3500 OR 00983500
bibdate: Sat Nov 1 19:57:00 MDT 2008
bibtimestamp: 2008.11.01 19:57:00 MDT
fjournal: ACM Transactions on Mathematical Software (TOMS)
keywords: error function; logarithm of the standard normal distribution; Normal distribution; normal integral
acknowledgement: ack-nhfb
articleno: 20
journalURL: https://dl.acm.org/loi/toms
rawauthor: Jean Marie Linhart
rawtitle: {Algorithm 885}: Computing the Logarithm of the Normal Distribution
abstract:
We present and compare three C functions to compute the logarithm of the cumulative standard normal distribution. The first is a new algorithm derived from Algorithm 304's calculation of the standard normal distribution via a series or continued fraction approximation, and it is good to the accuracy of the machine. The second is based on Algorithm 715's calculation of the standard normal distribution via rational Chebyshev approximation. This is related to, and an improvement on, the algorithm for the logarithm of the normal distribution available in the software package R. The third is a new and simple algorithm that uses the compiler's implementation of the error function, and complement of the error function, to compute the log of the normal distribution.

rawabstract:
We present and compare three C functions to compute the logarithm of the cumulative standard normal distribution. The first is a new algorithm derived from Algorithm 304's calculation of the standard normal distribution via a series or continued fraction approximation, and it is good to the accuracy of the machine. The second is based on Algorithm 715's calculation of the standard normal distribution via rational Chebyshev approximation. This is related to, and an improvement on, the algorithm for the logarithm of the normal distribution available in the software package R. The third is a new and simple algorithm that uses the compiler's implementation of the error function, and complement of the error function, to compute the log of the normal distribution.

entry:
@Article{Linhart:2009:ACL,
  author =       "Jean Marie Linhart",
  title =        "{Algorithm 885}: Computing the Logarithm of the Normal
                 Distribution",
  journal =      j-TOMS,
  volume =       "35",
  number =       "3",
  pages =        "20:1--20:10",
  month =        oct,
  year =         "2009",
  CODEN =        "ACMSCU",
  DOI =          "https://doi.org/10.1145/1391989.1391993",
  ISSN =         "0098-3500 (print), 1557-7295 (electronic)",
  ISSN-L =       "0098-3500",
  bibdate =      "Sat Nov 1 19:57:00 MDT 2008",
  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 =     "We present and compare three C functions to compute
                 the logarithm of the cumulative standard normal
                 distribution. The first is a new algorithm derived from
                 Algorithm 304's calculation of the standard normal
                 distribution via a series or continued fraction
                 approximation, and it is good to the accuracy of the
                 machine. The second is based on Algorithm 715's
                 calculation of the standard normal distribution via
                 rational Chebyshev approximation. This is related to,
                 and an improvement on, the algorithm for the logarithm
                 of the normal distribution available in the software
                 package R. The third is a new and simple algorithm that
                 uses the compiler's implementation of the error
                 function, and complement of the error function, to
                 compute the log of the normal distribution.",
  acknowledgement = ack-nhfb,
  articleno =    "20",
  fjournal =     "ACM Transactions on Mathematical Software (TOMS)",
  journal-URL =  "https://dl.acm.org/loi/toms",
  keywords =     "error function; logarithm of the standard normal
                 distribution; Normal distribution; normal integral",
}


bibtype: article
filename: elefunt.bib
label: Loskot:2009:PPA
author: P. Loskot and N. C. Beaulieu
journal: j-IEEE-TRANS-VEH-TECHNOL
volume: 58
number: 3
pages: 1269--1280
month: mar
monthnumber: 3
year: 2009
CODEN: ITUTAB
DOI: https://doi.org/10.1109/tvt.2008.926072
ISSN: 0018-9545 (print), 1939-9359 (electronic) OR 00189545 (print), 19399359 (electronic)
ISSNL: 0018-9545 OR 00189545
bibdate: Sat Dec 16 18:08:41 2017
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2017.12.16 18:08:41 ???
URL: http://ieeexplore.ieee.org/document/4529094/
fjournal: IEEE Transactions on Vehicular Technology
acknowledgement: ack-nhfb
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=25
rawauthor: P. Loskot and N. C. Beaulieu
title:
Prony and Polynomial Approximations for Evaluation of the Average Probability of Error Over Slow-Fading Channels

rawtitle:
{Prony} and Polynomial Approximations for Evaluation of the Average Probability of Error Over Slow-Fading Channels

entry:
@Article{Loskot:2009:PPA,
  author =       "P. Loskot and N. C. Beaulieu",
  title =        "{Prony} and Polynomial Approximations for Evaluation
                 of the Average Probability of Error Over Slow-Fading
                 Channels",
  journal =      j-IEEE-TRANS-VEH-TECHNOL,
  volume =       "58",
  number =       "3",
  pages =        "1269--1280",
  month =        mar,
  year =         "2009",
  CODEN =        "ITUTAB",
  DOI =          "https://doi.org/10.1109/tvt.2008.926072",
  ISSN =         "0018-9545 (print), 1939-9359 (electronic)",
  ISSN-L =       "0018-9545",
  bibdate =      "Sat Dec 16 18:08:41 2017",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
  URL =          "http://ieeexplore.ieee.org/document/4529094/",
  acknowledgement = ack-nhfb,
  fjournal =     "IEEE Transactions on Vehicular Technology",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=25",
}


bibtype: article
filename: elefunt.bib
label: Nagayama:2009:CGB
author: S. Nagayama and T. Sasao
title: Complexities of Graph-Based Representations for Elementary Functions
journal: j-IEEE-TRANS-COMPUT
volume: 58
number: 1
pages: 106--119
month: jan
monthnumber: 1
year: 2009
CODEN: ITCOB4
DOI: https://doi.org/10.1109/TC.2008.134
ISSN: 0018-9340 (print), 1557-9956 (electronic) OR 00189340 (print), 15579956 (electronic)
ISSNL: 0018-9340 OR 00189340
bibdate: Mon Jul 4 11:37:39 MDT 2011
bibtimestamp: 2011.07.04 11:37:39 MDT
URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4599569
fjournal: IEEE Transactions on Computers
acknowledgement: ack-nhfb
ajournal: IEEE Trans. Comput.
journalURL: http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12
rawauthor: S. Nagayama and T. Sasao
rawtitle: Complexities of Graph-Based Representations for Elementary Functions
abstract:
This paper analyzes complexities of decision diagrams for elementary functions such as polynomial, trigonometric, logarithmic, square root, and reciprocal functions. These real functions are converted into integer-valued functions by using fixed-point representation. This paper presents the numbers of nodes in decision diagrams representing the integer-valued functions. First, complexities of decision diagrams for polynomial functions are analyzed, since elementary functions can be approximated by polynomial functions. A theoretical analysis shows that binary moment diagrams (BMDs) have low complexity for polynomial functions. Second, this paper analyzes complexity of edge-valued binary decision diagrams (EVBDDs) for monotone functions, since many common elementary functions are monotone. It introduces a new class of integer functions, Mp-monotone increasing function, and derives an upper bound on the number of nodes in an EVBDD for the Mp-monotone increasing function. A theoretical analysis shows that EVBDDs have low complexity for Mp-monotone increasing functions. This paper also presents the exact number of nodes in the smallest EVBDD for the n-bit multiplier function, and a variable order for the smallest EVBDD.

keywords:
binary moment diagram; decision diagrams; edge-valued binary decision diagram; elementary function; elementary function approximation; fixed-point representation; general representations; graph-based representation; integer-valued function; monotone function; polynomial function; trees

rawabstract:
This paper analyzes complexities of decision diagrams for elementary functions such as polynomial, trigonometric, logarithmic, square root, and reciprocal functions. These real functions are converted into integer-valued functions by using fixed-point representation. This paper presents the numbers of nodes in decision diagrams representing the integer-valued functions. First, complexities of decision diagrams for polynomial functions are analyzed, since elementary functions can be approximated by polynomial functions. A theoretical analysis shows that binary moment diagrams (BMDs) have low complexity for polynomial functions. Second, this paper analyzes complexity of edge-valued binary decision diagrams (EVBDDs) for monotone functions, since many common elementary functions are monotone. It introduces a new class of integer functions, Mp-monotone increasing function, and derives an upper bound on the number of nodes in an EVBDD for the Mp-monotone increasing function. A theoretical analysis shows that EVBDDs have low complexity for Mp-monotone increasing functions. This paper also presents the exact number of nodes in the smallest EVBDD for the n-bit multiplier function, and a variable order for the smallest EVBDD.

entry:
@Article{Nagayama:2009:CGB,
  author =       "S. Nagayama and T. Sasao",
  title =        "Complexities of Graph-Based Representations for
                 Elementary Functions",
  journal =      j-IEEE-TRANS-COMPUT,
  volume =       "58",
  number =       "1",
  pages =        "106--119",
  month =        jan,
  year =         "2009",
  CODEN =        "ITCOB4",
  DOI =          "https://doi.org/10.1109/TC.2008.134",
  ISSN =         "0018-9340 (print), 1557-9956 (electronic)",
  ISSN-L =       "0018-9340",
  bibdate =      "Mon Jul 4 11:37:39 MDT 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 https://www.math.utah.edu/pub/tex/bib/ieeetranscomput.bib",
  URL =          "http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=4599569",
  abstract =     "This paper analyzes complexities of decision diagrams
                 for elementary functions such as polynomial,
                 trigonometric, logarithmic, square root, and reciprocal
                 functions. These real functions are converted into
                 integer-valued functions by using fixed-point
                 representation. This paper presents the numbers of
                 nodes in decision diagrams representing the
                 integer-valued functions. First, complexities of
                 decision diagrams for polynomial functions are
                 analyzed, since elementary functions can be
                 approximated by polynomial functions. A theoretical
                 analysis shows that binary moment diagrams (BMDs) have
                 low complexity for polynomial functions. Second, this
                 paper analyzes complexity of edge-valued binary
                 decision diagrams (EVBDDs) for monotone functions,
                 since many common elementary functions are monotone. It
                 introduces a new class of integer functions,
                 Mp-monotone increasing function, and derives an upper
                 bound on the number of nodes in an EVBDD for the
                 Mp-monotone increasing function. A theoretical analysis
                 shows that EVBDDs have low complexity for Mp-monotone
                 increasing functions. This paper also presents the
                 exact number of nodes in the smallest EVBDD for the
                 n-bit multiplier function, and a variable order for the
                 smallest EVBDD.",
  acknowledgement = ack-nhfb,
  ajournal =     "IEEE Trans. Comput.",
  fjournal =     "IEEE Transactions on Computers",
  journal-URL =  "http://ieeexplore.ieee.org/xpl/RecentIssue.jsp?punumber=12",
  keywords =     "binary moment diagram; decision diagrams; edge-valued
                 binary decision diagram; elementary function;
                 elementary function approximation; fixed-point
                 representation; general representations; graph-based
                 representation; integer-valued function; monotone
                 function; polynomial function; trees",
}


bibtype: book
filename: elefunt.bib
label: Oldham:2009:AF
editor: Keith B. Oldham and Jan Myland and Jerome Spanier
title: An Atlas of Functions: With Equator, the Atlas Function Calculator
publisher: pub-SV
address: pub-SV:adr
edition: Second
pages: xi + 748
year: 2009
DOI: https://doi.org/10.1007/978-0-387-48807-3
LCCN: QA331 .S685 2009
bibdate: Fri Aug 31 16:20:13 MDT 2007
bibtimestamp: 2007.08.31 16:20:13 MDT
subject: Functions
acknowledgement: ack-nhfb
raweditor: Keith B. Oldham and Jan Myland and Jerome Spanier
rawtitle: An Atlas of Functions: With Equator, the Atlas Function Calculator
ISBN:
0-387-48807-3 (softcover), 0-387-48806-5 (hardcover) OR 0387488073 (softcover), 0387488065 (hardcover)

ISBN13:
978-0-387-48807-3 (softcover), 978-0-387-48806-6 (hardcover) OR 9780387488073 (softcover), 9780387488066 (hardcover)

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

TOC:
Front Matter / i--xi General Considerations / 1--11 The Constant Function $c$ / 13--19 The Factorial Function $n!$ / 21--27 The Zeta Numbers and Related Functions / 29--38 The Bernoulli Numbers $B_n$ / 39--44 The Euler Numbers $E_n$ / 45--48 The Binomial Coefficients $binom nu m$ / 49--56 The Linear Function $b x + c$ and Its Reciprocal / 57--65 Modifying Functions / 67--74 The Heaviside $u(x - a)$ And Dirac $delta (x - a)$ Functions / 75--80 The Integer Powers $x^n$ and $(b x + c)^n$ / 81--93 The Square-Root Function $sqrt b x + c$ and Its Reciprocal / 95--102 The Noninteger Powers $x^nu $ / 103--112 The Semielliptic Function $(b / a)sqrt a^2 - x^2$ and Its Reciprocal / 113--120 The Semihyperbolic Functions $(b / a) sqrt x^2 pm a^2$ and Their Reciprocals / 121--130 The Quadratic Function $a x^2 + b x + c$ and Its Reciprocal / 131--138 The Cubic Function $x^3 + a x^2 + b x + c$ / 139--146 Polynomial Functions / 147--158 The Pochhammer Polynomials $(x)_n$ / 159--174 The Bernoulli Polynomials $B_n(x)$ / 175--180 The Euler Polynomials $E_n(x)$ / 181--186 The Legendre Polynomials $P_n(x)$ / 187--196 The Chebyshev Polynomials $T_n(x)$ and $U_n(x)$ / 197--208 The Laguerre Polynomials $L_n(x)$ / 209--216 The Hermite Polynomials $H_n(x)$ / 217--227 The Logarithmic Function $ln (x)$ / 229--239 The Exponential Function $exp (pm x)$ / 241--253 Exponentials of Powers $exp (pm x^nu )$ / 255--267 The Hyperbolic Cosine $cosh (x)$ and Sine $sinh (x)$ Functions / 269--279 The Hyperbolic Secant $sech (x)$ and Cosecant $csch (x)$ Functions / 281--288 The Hyperbolic Tangent $tanh (x)$ and Cotangent $coth (x)$ Functions / 289--296 The Inverse Hyperbolic Functions / 297--307 The Cosine $cos (x)$ and Sine $sin (x)$ Functions / 309--328 The Secant $sec (x)$ and Cosecant $csc (x)$ Functions / 329--338 The Tangent $tan (x)$ and Cotangent $cot (x)$ Functions / 339--350 The Inverse Circular Functions / 351--366 Periodic Functions / 367--374 The Exponential Integrals $Ei (x)$ and $Ein (x)$ / 375--383 Sine and Cosine Integrals / 385--394 The Fresnel Integrals $C(x)$ and $S(x)$ / 395--404 The Error Function $erf (x)$ and Its Complement $erfc (x)$ / 405--415 The $exp (x)erfc (sqrt x)$ and Related Functions / 417--426 Dawson's Integral $daw (x)$ / 427--433 The Gamma Function $Gamma (nu )$ / 435--448 The Digamma Function $psi (nu )$ / 449--460 The Incomplete Gamma Functions / 461--470 The Parabolic Cylinder Function $D_nu (x)$ / 471--484 The Kummer Function $M(a, c, x)$ / 485--496 The Tricomi Function $U(a, c, x)$ / 497--506 The Modified Bessel Functions $I_n(x)$ of Integer Order / 507--517 The Modified Bessel Function $I_nu (x)$ of Arbitrary Order / 519--526 The Macdonald Function $K_nu (x)$ / 527--536 The Bessel Functions $J_n(x)$ of Integer Order / 537--552 The Bessel Function $J_nu (x)$ of Arbitrary Order / 553--565 The Neumann Function $Y_nu (x)$ / 567--576 The Kelvin Functions / 577--584 The Airy Functions $Ai (x)$ and $Bi (x)$ / 585--592 The Struve Function $h_nu (x)$ / 593--601 The Incomplete Beta Function $B(nu , mu , x)$ / 603--609 The Legendre Functions $P_nu (x)$ and $Q_nu (x)$ / 611--626 The Gauss Hypergeometric Function $F(a, b, c, x)$ / 627--636 The Complete Elliptic Integrals $K(k)$ and $E(k)$ / 637--651 The Incomplete Elliptic Integrals $F(k, phi )$ and $E(k, phi )$ / 653--669 The Jacobian Elliptic Functions / 671--684 The Hurwitz Function $zeta (nu , u)$ / 685--695 Back Matter / 697--748

rawTOC:
Front Matter / i--xi \\ General Considerations / 1--11 \\ The Constant Function $c$ / 13--19 \\ The Factorial Function $n!$ / 21--27 \\ The Zeta Numbers and Related Functions / 29--38 \\ The Bernoulli Numbers $B_n$ / 39--44 \\ The Euler Numbers $E_n$ / 45--48 \\ The Binomial Coefficients $\binom{\nu}{m}$ / 49--56 \\ The Linear Function $b x + c$ and Its Reciprocal / 57--65 \\ Modifying Functions / 67--74 \\ The Heaviside $u(x - a)$ And Dirac $\delta(x - a)$ Functions / 75--80 \\ The Integer Powers $x^n$ and $(b x + c)^n$ / 81--93 \\ The Square-Root Function $\sqrt{b x + c}$ and Its Reciprocal / 95--102 \\ The Noninteger Powers $x^\nu$ / 103--112 \\ The Semielliptic Function $(b / a)\sqrt{a^2 - x^2}$ and Its Reciprocal / 113--120 \\ The Semihyperbolic Functions $(b / a) \sqrt{x^2 \pm a^2}$ and Their Reciprocals / 121--130 \\ The Quadratic Function $a x^2 + b x + c$ and Its Reciprocal / 131--138 \\ The Cubic Function $x^3 + a x^2 + b x + c$ / 139--146 \\ Polynomial Functions / 147--158 \\ The Pochhammer Polynomials $(x)_n$ / 159--174 \\ The Bernoulli Polynomials $B_n(x)$ / 175--180 \\ The Euler Polynomials $E_n(x)$ / 181--186 \\ The Legendre Polynomials $P_n(x)$ / 187--196 \\ The Chebyshev Polynomials $T_n(x)$ and $U_n(x)$ / 197--208 \\ The Laguerre Polynomials $L_n(x)$ / 209--216 \\ The Hermite Polynomials $H_n(x)$ / 217--227 \\ The Logarithmic Function $\ln(x)$ / 229--239 \\ The Exponential Function $\exp(\pm x)$ / 241--253 \\ Exponentials of Powers $\exp(\pm x^\nu)$ / 255--267 \\ The Hyperbolic Cosine $\cosh(x)$ and Sine $\sinh(x)$ Functions / 269--279 \\ The Hyperbolic Secant $\sech(x)$ and Cosecant $\csch(x)$ Functions / 281--288 \\ The Hyperbolic Tangent $\tanh(x)$ and Cotangent $\coth(x)$ Functions / 289--296 \\ The Inverse Hyperbolic Functions / 297--307 \\ The Cosine $\cos(x)$ and Sine $\sin(x)$ Functions / 309--328 \\ The Secant $\sec(x)$ and Cosecant $\csc(x)$ Functions / 329--338 \\ The Tangent $\tan(x)$ and Cotangent $\cot(x)$ Functions / 339--350 \\ The Inverse Circular Functions / 351--366 \\ Periodic Functions / 367--374 \\ The Exponential Integrals $\Ei(x)$ and $\Ein(x)$ / 375--383 \\ Sine and Cosine Integrals / 385--394 \\ The Fresnel Integrals $C(x)$ and $S(x)$ / 395--404 \\ The Error Function $\erf(x)$ and Its Complement $\erfc(x)$ / 405--415 \\ The $\exp(x)\erfc(\sqrt{x})$ and Related Functions / 417--426 \\ Dawson's Integral $\daw(x)$ / 427--433 \\ The Gamma Function $\Gamma(\nu)$ / 435--448 \\ The Digamma Function $\psi(\nu)$ / 449--460 \\ The Incomplete Gamma Functions / 461--470 \\ The Parabolic Cylinder Function $D_\nu(x)$ / 471--484 \\ The Kummer Function $M(a, c, x)$ / 485--496 \\ The Tricomi Function $U(a, c, x)$ / 497--506 \\ The Modified Bessel Functions $I_n(x)$ of Integer Order / 507--517 \\ The Modified Bessel Function $I_\nu(x)$ of Arbitrary Order / 519--526 \\ The Macdonald Function $K_\nu(x)$ / 527--536 \\ The Bessel Functions $J_n(x)$ of Integer Order / 537--552 \\ The Bessel Function $J_\nu(x)$ of Arbitrary Order / 553--565 \\ The Neumann Function $Y_\nu(x)$ / 567--576 \\ The Kelvin Functions / 577--584 \\ The Airy Functions $\Ai(x)$ and $\Bi(x)$ / 585--592 \\ The Struve Function $h_\nu(x)$ / 593--601 \\ The Incomplete Beta Function $B(\nu, \mu, x)$ / 603--609 \\ The Legendre Functions $P_\nu(x)$ and $Q_\nu(x)$ / 611--626 \\ The Gauss Hypergeometric Function $F(a, b, c, x)$ / 627--636 \\ The Complete Elliptic Integrals $K(k)$ and $E(k)$ / 637--651 \\ The Incomplete Elliptic Integrals $F(k, \phi)$ and $E(k, \phi)$ / 653--669 \\ The Jacobian Elliptic Functions / 671--684 \\ The Hurwitz Function $\zeta(\nu, u)$ / 685--695 \\ Back Matter / 697--748

entry:
@Book{Oldham:2009:AF,
  editor =       "Keith B. Oldham and Jan Myland and Jerome Spanier",
  title =        "An Atlas of Functions: With Equator, the Atlas
                 Function Calculator",
  publisher =    pub-SV,
  address =      pub-SV:adr,
  edition =      "Second",
  pages =        "xi + 748",
  year =         "2009",
  DOI =          "https://doi.org/10.1007/978-0-387-48807-3",
  ISBN =         "0-387-48807-3 (softcover), 0-387-48806-5 (hardcover)",
  ISBN-13 =      "978-0-387-48807-3 (softcover), 978-0-387-48806-6
                 (hardcover)",
  LCCN =         "QA331 .S685 2009",
  bibdate =      "Fri Aug 31 16:20:13 MDT 2007",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/canjstat.bib;
                 https://www.math.utah.edu/pub/tex/bib/elefunt.bib;
                 z3950.loc.gov:7090/Voyager",
  price =        "US\$129.95",
  acknowledgement = ack-nhfb,
  subject =      "Functions",
  tableofcontents = "Front Matter / i--xi \\
                 General Considerations / 1--11 \\
                 The Constant Function $c$ / 13--19 \\
                 The Factorial Function $n!$ / 21--27 \\
                 The Zeta Numbers and Related Functions / 29--38 \\
                 The Bernoulli Numbers $B_n$ / 39--44 \\
                 The Euler Numbers $E_n$ / 45--48 \\
                 The Binomial Coefficients $\binom{\nu}{m}$ / 49--56 \\
                 The Linear Function $b x + c$ and Its Reciprocal /
                 57--65 \\
                 Modifying Functions / 67--74 \\
                 The Heaviside $u(x - a)$ And Dirac $\delta(x - a)$
                 Functions / 75--80 \\
                 The Integer Powers $x^n$ and $(b x + c)^n$ / 81--93 \\
                 The Square-Root Function $\sqrt{b x + c}$ and Its
                 Reciprocal / 95--102 \\
                 The Noninteger Powers $x^\nu$ / 103--112 \\
                 The Semielliptic Function $(b / a)\sqrt{a^2 - x^2}$ and
                 Its Reciprocal / 113--120 \\
                 The Semihyperbolic Functions $(b / a) \sqrt{x^2 \pm
                 a^2}$ and Their Reciprocals / 121--130 \\
                 The Quadratic Function $a x^2 + b x + c$ and Its
                 Reciprocal / 131--138 \\
                 The Cubic Function $x^3 + a x^2 + b x + c$ / 139--146
                 \\
                 Polynomial Functions / 147--158 \\
                 The Pochhammer Polynomials $(x)_n$ / 159--174 \\
                 The Bernoulli Polynomials $B_n(x)$ / 175--180 \\
                 The Euler Polynomials $E_n(x)$ / 181--186 \\
                 The Legendre Polynomials $P_n(x)$ / 187--196 \\
                 The Chebyshev Polynomials $T_n(x)$ and $U_n(x)$ /
                 197--208 \\
                 The Laguerre Polynomials $L_n(x)$ / 209--216 \\
                 The Hermite Polynomials $H_n(x)$ / 217--227 \\
                 The Logarithmic Function $\ln(x)$ / 229--239 \\
                 The Exponential Function $\exp(\pm x)$ / 241--253 \\
                 Exponentials of Powers $\exp(\pm x^\nu)$ / 255--267 \\
                 The Hyperbolic Cosine $\cosh(x)$ and Sine $\sinh(x)$
                 Functions / 269--279 \\
                 The Hyperbolic Secant $\sech(x)$ and Cosecant
                 $\csch(x)$ Functions / 281--288 \\
                 The Hyperbolic Tangent $\tanh(x)$ and Cotangent
                 $\coth(x)$ Functions / 289--296 \\
                 The Inverse Hyperbolic Functions / 297--307 \\
                 The Cosine $\cos(x)$ and Sine $\sin(x)$ Functions /
                 309--328 \\
                 The Secant $\sec(x)$ and Cosecant $\csc(x)$ Functions /
                 329--338 \\
                 The Tangent $\tan(x)$ and Cotangent $\cot(x)$ Functions
                 / 339--350 \\
                 The Inverse Circular Functions / 351--366 \\
                 Periodic Functions / 367--374 \\
                 The Exponential Integrals $\Ei(x)$ and $\Ein(x)$ /
                 375--383 \\
                 Sine and Cosine Integrals / 385--394 \\
                 The Fresnel Integrals $C(x)$ and $S(x)$ / 395--404 \\
                 The Error Function $\erf(x)$ and Its Complement
                 $\erfc(x)$ / 405--415 \\
                 The $\exp(x)\erfc(\sqrt{x})$ and Related Functions /
                 417--426 \\
                 Dawson's Integral $\daw(x)$ / 427--433 \\
                 The Gamma Function $\Gamma(\nu)$ / 435--448 \\
                 The Digamma Function $\psi(\nu)$ / 449--460 \\
                 The Incomplete Gamma Functions / 461--470 \\
                 The Parabolic Cylinder Function $D_\nu(x)$ / 471--484
                 \\
                 The Kummer Function $M(a, c, x)$ / 485--496 \\
                 The Tricomi Function $U(a, c, x)$ / 497--506 \\
                 The Modified Bessel Functions $I_n(x)$ of Integer Order
                 / 507--517 \\
                 The Modified Bessel Function $I_\nu(x)$ of Arbitrary
                 Order / 519--526 \\
                 The Macdonald Function $K_\nu(x)$ / 527--536 \\
                 The Bessel Functions $J_n(x)$ of Integer Order /
                 537--552 \\
                 The Bessel Function $J_\nu(x)$ of Arbitrary Order /
                 553--565 \\
                 The Neumann Function $Y_\nu(x)$ / 567--576 \\
                 The Kelvin Functions / 577--584 \\
                 The Airy Functions $\Ai(x)$ and $\Bi(x)$ / 585--592 \\
                 The Struve Function $h_\nu(x)$ / 593--601 \\
                 The Incomplete Beta Function $B(\nu, \mu, x)$ /
                 603--609 \\
                 The Legendre Functions $P_\nu(x)$ and $Q_\nu(x)$ /
                 611--626 \\
                 The Gauss Hypergeometric Function $F(a, b, c, x)$ /
                 627--636 \\
                 The Complete Elliptic Integrals $K(k)$ and $E(k)$ /
                 637--651 \\
                 The Incomplete Elliptic Integrals $F(k, \phi)$ and
                 $E(k, \phi)$ / 653--669 \\
                 The Jacobian Elliptic Functions / 671--684 \\
                 The Hurwitz Function $\zeta(\nu, u)$ / 685--695 \\
                 Back Matter / 697--748",
}


bibtype: article
filename: elefunt.bib
label: Opps:2009:RFA
author: Sheldon B. Opps and Nasser Saad and H. M. Srivastava
journal: j-APPL-MATH-COMP
volume: 207
number: 2
pages: 545--558
day: 15
month: jan
monthnumber: 1
year: 2009
CODEN: AMHCBQ
ISSN: 0096-3003 (print), 1873-5649 (electronic) OR 00963003 (print), 18735649 (electronic)
ISSNL: 0096-3003 OR 00963003
bibdate: Fri Sep 3 10:53:24 MDT 2010
bibtimestamp: 2010.09.03 10:53:24 MDT
fjournal: Applied Mathematics and Computation
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00963003
rawauthor: Sheldon B. Opps and Nasser Saad and H. M. Srivastava
title:
Recursion formulas for Appell's hypergeometric function $ F_2 $ with some applications to radiation field problems

bibsource:
http://www.sciencedirect.com/science/journal/00963003; https://www.math.utah.edu/pub/tex/bib/applmathcomput2005.bib; https://www.math.utah.edu/pub/tex/bib/elefunt.bib

rawtitle:
Recursion formulas for {Appell}'s hypergeometric function {$ F_2 $} with some applications to radiation field problems

entry:
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  author =       "Sheldon B. Opps and Nasser Saad and H. M. Srivastava",
  title =        "Recursion formulas for {Appell}'s hypergeometric
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bibtype: article
filename: elefunt.bib
label: Paris:2009:HPE
author: R. B. Paris
title: High-precision evaluation of the Bessel functions via Hadamard series
journal: j-J-COMPUT-APPL-MATH
volume: 224
number: 1
pages: 84--100
day: 1
month: feb
monthnumber: 2
year: 2009
CODEN: JCAMDI
ISSN: 0377-0427 (print), 1879-1778 (electronic) OR 03770427 (print), 18791778 (electronic)
ISSNL: 0377-0427 OR 03770427
bibdate: Sat Feb 25 13:13:29 MST 2017
bibtimestamp: 2017.02.25 13:13:29 MST
URL: http://www.sciencedirect.com/science/article/pii/S0377042708001799
fjournal: Journal of Computational and Applied Mathematics
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03770427
rawauthor: R. B. Paris
rawtitle: High-precision evaluation of the {Bessel} functions via {Hadamard} series
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  number =       "1",
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bibtype: mastersthesis
filename: elefunt.bib
label: Pearson:2009:CHF
author: J. Pearson
title: Computation of hypergeometric functions
type: Master's thesis
school: Oxford University
address: Oxford, UK
pages: 84--100
year: 2009
bibdate: Thu Dec 01 09:05:26 2011
bibsource: https://www.math.utah.edu/pub/tex/bib/elefunt.bib
bibtimestamp: 2011.12.01 09:05:26 ???
acknowledgement: ack-nhfb
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rawtitle: Computation of hypergeometric functions
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  title =        "Computation of hypergeometric functions",
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  address =      "Oxford, UK",
  year =         "2009",
  bibdate =      "Thu Dec 01 09:05:26 2011",
  bibsource =    "https://www.math.utah.edu/pub/tex/bib/elefunt.bib",
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bibtype: article
filename: elefunt.bib
label: Talman:2009:NSC
author: J. D. Talman
title: NumSBT: a subroutine for calculating spherical Bessel transforms numerically
journal: j-COMP-PHYS-COMM
volume: 180
number: 2
pages: 332--338
month: feb
monthnumber: 2
year: 2009
CODEN: CPHCBZ
DOI: https://doi.org/10.1016/j.cpc.2008.10.003
ISSN: 0010-4655 (print), 1879-2944 (electronic) OR 00104655 (print), 18792944 (electronic)
ISSNL: 0010-4655 OR 00104655
bibdate: Mon Feb 13 23:42:39 MST 2012
bibtimestamp: 2012.02.13 23:42:39 MST
URL: http://www.sciencedirect.com/science/article/pii/S0010465508003329
fjournal: Computer Physics Communications
acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/00104655
rawauthor: J. D. Talman
rawtitle: {NumSBT}: a subroutine for calculating spherical {Bessel} transforms numerically
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bibtype: article
filename: elefunt.bib
label: Temme:2009:AER
author: Nico M. Temme and Vladimir Varlamov
title: Asymptotic expansions for Riesz fractional derivatives of Airy functions and applications
journal: j-J-COMPUT-APPL-MATH
volume: 232
number: 2
pages: 201--215
day: 15
month: oct
monthnumber: 10
year: 2009
CODEN: JCAMDI
ISSN: 0377-0427 (print), 1879-1778 (electronic) OR 03770427 (print), 18791778 (electronic)
ISSNL: 0377-0427 OR 03770427
bibdate: Sat Feb 25 13:24:17 MST 2017
bibtimestamp: 2017.02.25 13:24:17 MST
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acknowledgement: ack-nhfb
journalURL: http://www.sciencedirect.com/science/journal/03770427
rawauthor: Nico M. Temme and Vladimir Varlamov
rawtitle: Asymptotic expansions for {Riesz} fractional derivatives of {Airy} functions and applications
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