[Search][Subject Index][MathMap][Tour][Help!]

# 33: Special functions

## Introduction

Special functions are just that: specialized functions beyond the familiar trigonometric or exponential functions. The ones studied (hypergeometric functions, orthogonal polynomials, and so on) arise very naturally in areas of analysis, number theory, Lie groups, and combinatorics. Very detailed information is often available.

Among the functions studied: Trigonometric functions, Exponential functions, Hyperbolic functions, Error functions, Elliptic integrals, Gamma functions, Bessel functions, Fresnel integrals, Airy functions, Kelvin functions, Pochhammer's symbols

## Applications and related fields

33-XX deals with the properties of functions as functions per se. For aspects of combinatorics, see 05AXX; for number-theoretic aspects, see 11-XX; for representation theory, see 22EXX; for orthogonal functions, see also 42CXX. For numerical computations of the special functions, see 65-XX. the data used for drawing the map are limited to papers since 1991: prior to that year, there was only one subdivision, 33A.

## Subfields

• 33B: Elementary classical functions
• 33C: Hypergeometric functions
• 33D: Basic hypergeometric functions
• 33E: Other special functions
• 33F: Computational aspects [new in 2000]
These subfields were only introduced in 1991; prior to that date all articles were classed in section 33A.

Browse all (old) classifications for this area at the AMS.

## Textbooks, reference works, and tutorials

Temme, Nico M. "Special functions. An introduction to the classical functions of mathematical physics", John Wiley & Sons, Inc., New York, 1996 ISBN 0-471-11313-1

Magnus, Wilhelm; Oberhettinger, Fritz; Soni, Raj Pal; "Formulas and theorems for the special functions of mathematical physics", Die Grundlehren der mathematischen Wissenschaften, Band 52 Springer-Verlag New York, Inc., New York 1966 508 pp.

Jerome Spanier, Keith B. Oldham, "An atlas of functions", Hemisphere Pub. Corp., Washington, 1987, 700pp. ISBN 0-891-16573-8

Abramowitz, M. and Stegun, C.A. (Ed.). "Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables", 9th printing, 1972, New York: Dover.

Gradshteyn and Ryzhik; Prudnikov et al

Neville, Eric Harold: "Elliptic functions: a primer" Pergamon Press, Oxford-New York-Toronto, Ont., 1971. 198 pp. MR58#17242

There is a special-functions mailing list.

## Software and tables

• Elementary and special functions software
• The Maple share library orbitals includes procedures for the computation of real and complex spherical solid and surface harmonics.
• Math Functions packages for Mathematica, versions 2.2 and 3.0.
• There are also Hypergeometric Series packages for Mathematica.