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[Texts]## 33: Special functions |

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

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.

- 33B: Elementary classical functions
- 33C: Hypergeometric functions
- 33D: Basic hypergeometric functions
- 33E: Other special functions
- 33F: Computational aspects [new in 2000]

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

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.

- Elementary and special functions software
- There is a large formulary for hypergeometric functions.
- 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.

- There is a SIAM Activity Group on Orthogonal Polynomials and Special Functions. Their web site points to the OP-SF NET (electronic newsletter), preprint servers, subject classification information, and other professional concerns.
- Here are the AMS and Goettingen resource pages for area 33.

- What are the Legendre polynomials?
- What are the Chebyshev polynomials and what are they good for?
- Use of the Chebyshev polynomials (or other orthogonal families) for approximating other functions.
- Computing the Chebyshev polynomials nonsequentially.
- Families of polynomials which commute under composition must be either powers or the Chebyshev polynomials.
- Integral definitions of the Gamma function.
- Estimating (x+0.5)!/x! with the Gamma function.
- Extending the domain of the subfactorial function (hypergeometric functions)
- A 2nd order linear equation. After the fact we have learned: (a)about Bessel functions (2)to use a symbolic algebra program to solve differential equations.
- Finding an orthogonal family of functions with vanishing derivatives at endpoints.
- Computing elliptic integrals with the arithmetic-geometric mean (See also references in PI bibliography.
- What are lemniscatic functions? (Defined with elliptic integrals.)
- References on the Lambert W-function (defined by x=W(x)*exp(W(x)) ).
- Relating the complex trigonometric and exponential functions.
- Formal definition of the sine function (via integrals) and derivation of some of its properties.
- SPHEREPACK, a set of FORTRAN programs that handles spherical harmonic expansions.
- Pointer to information about (and pictures of) spherical harmonics.
- Miscellaneous typos and errors in standard reference materials.

Last modified 1999/05/12 by Dave Rusin. Mail: feedback@math-atlas.org