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**D. W. Lozier and F. W. J. Olver**

Mathematical and Computational Sciences Division

National Institute of Standards and Technology

Gaithersburg, Md 20899-8910

Institute for Physical Science and Technology

University of Maryland

College Park, MD 20742

1991 Mathematics Subject Classification: Primary 65D20; Secondary 33-00

Higher transcendental functions continue to play varied and
important roles in investigations by engineers, mathematicians,
scientists and statisticians.
The purpose of this paper is to assist in locating useful approximations
and software for the numerical generation of these functions, and to
offer some suggestions for future developments in this field.

March 1994

December 2000

December 2000

1991 Mathematics Subject Classification: Primary 65D20;
Secondary 33-00.

The research of the second author has been supported by NSF Grant CCR 89-14933.

The December 2000 revision of this paper is available in hard copy from the authors
and on the Web at http://math.nist.gov/nesf/.

Preface

The article printed in this report will appear in Walter Gautschi (ed.),
*Mathematics
of Computation 1943-1993: A Half-Century of Computational Mathematics*,
Proceedings of Symposia in
Applied Mathematics, American Mathematical Society,
Providence, Rhode Island 02940.
This report is intended for limited distribution only until
the primary publication appears in print.

May 1994

Additional copies of this report have been made for distribution to Digital Library of Mathematical Functions Project Participants (editors, associate editors, authors, validators, and support staff). No changes have been made in the body of the report. The exact bibliographic reference for the published article is [LO94]. A revision is in preparation and will be provided to DLMF Project Participants when ready. It will include references that have appeared in the literature since 1993.

May 2000

This report has been updated through 1999, resulting in an approximate 15% increase in the number of references and a modest expansion of the classification scheme in §4 and §5. The software packages, libraries and systems, described in §3 and cross-referenced in §4 and §5, were re-examined. Maple and Mathematica (§3.4.3 and §3.4.5) were found to have added considerable support for special functions, and three new libraries (§3.2.1, §3.2.3 and §3.3.3) were included. For the Web version, see http://math.nist.gov/nesf.

December 2000

- Contents
- 1 Introduction
- 2 Mathematical Developments
- 3 Packages, Libraries and Systems
- 3.1 Software Packages
- 3.2 Intermediate Libraries
- 3.3 Comprehensive Libraries
- 3.4 Interactive Systems

- 4 Functions of One Variable
- 4.1 Airy Functions
- 4.2 Error Functions, Dawson's Integral, Fresnel Integrals, Goodwin-Staton Integral
- 4.3 Exponential Integrals, Logarithmic Integral, Sine and Cosine Integrals
- 4.4 Gamma, Psi, and Polygamma Functions
- 4.5 Landau Density and Distribution Functions
- 4.6 Polylogarithms, Clausen Integral
- 4.7 Zeta Function
- 4.8 Additional Functions of One Variable

- 5 Functions of Two or More Variables
- 5.1 Bessel Functions
- 5.2 Coulomb Wave Functions
- 5.3 Elliptic Integrals and Functions
- 5.4 Fermi-Dirac, Bose-Einstein, and Debye Integrals
- 5.5 Hypergeometric and Confluent Hypergeometric Functions
- 5.6 Incomplete Bessel Functions, Incomplete Beta Function
- 5.7 Incomplete Gamma Functions, Generalized Exponential Integrals
- 5.8 Legendre Functions and Associated Legendre Functions
- 5.9 Mathieu, Lamé, and Spheroidal Wave Functions
- 5.10 Orthogonal Polynomials
- 5.11 Polylogarithms (Generalized)
- 5.12 Struve and Anger-Weber Functions
- 5.13 Weber Parabolic Cylinder Functions
- 5.14 Zeta Function (Generalized)
- 5.15 Additional Functions of Two or More Variables

- 6 Testing and Library Construction
- 7 Future Trends
- Acknowledgments
- A Note on the Reference Acronyms
- Bibliography