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# Numerical Evaluation of Special Functions

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

** 5. Functions of Two or More Variables
**

As in § 4, an indication is made
of the programming language where applicable
and special note is made of references that include surveys.
Libraries and interactive systems are listed separately,
and similar remarks apply about the inclusiveness of the subsections.

** 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 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)
. **

### Abstract:

*This document is an excerpt from the current hypertext version of
an article that appeared in Walter Gautschi (ed.),
** Mathematics of Computation 1943--1993: A Half-Century of
Computational Mathematics*, Proceedings of Symposia in
Applied Mathematics 48, American Mathematical Society,
Providence, RI 02940, 1994.
The symposium was held at the University of British Columbia
August 9--13, 1993, in honor of the fiftieth anniversary of
the journal * Mathematics of Computation*.
The original abstract follows.

*
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.
*

Applied and Computational Mathematics Division,
National Institute of Standards and Technology,
Gaithersburg, Md 20899

* E-mail address: dlozier@nist.gov*

Institute for Physical Science and Technology,
University of Maryland,
College Park, MD 20742

* E-mail address: olver@bessel.umd.edu*

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

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

*Daniel W Lozier *

Fri Apr 7 13:58:29 EDT 1995