Special Functions

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special functions

[′spesh·əl ′fəŋk·shənz]
The various families of solution functions corresponding to cases of the hypergeometric equation or functions used in the equation's study, such as the gamma function.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

Special Functions


in mathematics, functions belonging to various special classes that are encountered particularly often in solving problems in mathematical physics. The principal special functions are solutions of second-order linear differential equations with variable coefficients. Examples of important special functions are the hypergeometric functions, cylindrical functions, spherical functions, solid spherical harmonics, Lamé functions, and Mathieu functions. Transcendental functions that cannot be expressed in terms of elementary functions are sometimes also called special functions. Such transcendental functions include elliptic functions, the gamma function, the zeta function, the logarithmic integral, and the error function.


Smirnov, V. I. Kurs vysshei matematiki, 8th ed., vol. 3, part 2. Moscow, 1969.
Whittaker, E. T., and G. N. Watson. Kurs sovremennogo analiza, 2nd ed., part 2. Moscow, 1963. (Translated from English.)
Jahnke, E , F. Emde, and F. Lósch. Spetsial’nye funktsii: Formuly,grafiki, mblitsy, 2nd ed. Moscow, 1968. (Translated from German; contains bibliography. )
The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.
References in classic literature ?
And the art of payment has the special function of giving pay: but we do not confuse this with other arts, any more than the art of the pilot is to be confused with the art of medicine, because the health of the pilot may be improved by a sea voyage.
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There Gautschi developed a number of computer algorithms for evaluating special functions, for example, for the gamma and incomplete gamma functions, and Bessel functions of the first kind.
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Paneva-Konovska presents readers with a comprehensive examination of the use of Bessel and Mittag-Leffler functions within the fields of mathematics and the sciences through the study of enumerable families of different classes of special functions. The author covers Bessel and associated functions, generating functions of Bessel and associated Bessel functions, convergence of series in Bessel functions, and a wide variety of related subjects.
Recent advances in orthogonal polynomials, special functions, and their applications; proceedings.

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