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	ABOUT: [Introduction][History][Related areas][Subfields] POINTERS: [Texts][Software][Web links][Selected topics here] 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

History

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]

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.

Other web sites with this focus

• 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.

Selected topics at this site

• What are the Legendre polynomials?
• Rodrigues formulae for Laguerre 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.
• Code for computing values of the gamma function
• Use generating function to evaluate Beta function definite integrals
• Use of incomplete Beta function to predict population proportions from a sample
• Meijer G functions and the incomplete gamma function
• Raabe's formula (integral of log(Gamma) )
• Generalization of binomial coefficient (Gamma, q-hypergeometric functions)
• Generalizations of the Riemann zeta function: Polygamma function, Hurwitz zeta, etc
• Reducing relations among Appell hypergeometric functions
• Numerical evaluation of complex hypergeometric series
• Extending the domain of the subfactorial function (hypergeometric functions)
• Whittaker functions (defined by a second-order linear ODE)
• 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.
• Formulas useful for numerical evaluation of Bessel functions
• Connections among Bessel, Hankel, Struve, Anger, Weber, Kelvin, and Airy functions (each a solution to a second-order linear ODE)
• Numerical evaluation of the Bessel functions
• Characterization of the Airy functions Ai and Bi
• Finding an orthogonal family of functions with vanishing derivatives at endpoints.
• Numerical evaluation of Jacobi elliptic functions
• Functions with an addition formula (F(x+y)=P(F(x),F(y)) P a polynomial) are elliptic functions
• Algebraic relations satisfied by elliptic functions
• Computing elliptic integrals with the arithmetic-geometric mean (See also references in PI bibliography.
• Algorithms to compute erf
• Summary of Fresnel integrals
• What are lemniscatic functions? (Defined with elliptic integrals.)
• References on the Lambert W-function (defined by x=W(x)*exp(W(x)) ).
• Summary of CORDIC algorithm for computing trigonometric functions
• Numerical calculation of arctangent function
• Generalizing trig identities to cover functions F(x)=exp(alpha*x) with alpha some other root of unity
• Relating the complex trigonometric and exponential functions.
• Formal definition of the sine function (via integrals) and derivation of some of its properties.
• Fortran 90 library for computation of special functions available
• Integral of product of spherical harmonics / Legendre Functions
• 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.