A special function is a named function that arises repeatedly in mathematics, mathematical physics, or related applications. The scope of the term varies among references.

Special functions arise as solutions of differential equations, as definite integrals, or as sums of series. The Airy functions and solve the Airy differential equation . The error function is proportional to the integral of from to . The Riemann zeta function is given by the Dirichlet series for .

Standard references group the gamma function with relatives such as the beta function and digamma function. Other familiar groups include Bessel functions, hypergeometric functions, orthogonal polynomials, and the related families of elliptic functions and elliptic integrals (Abramowitz and Stegun 1972).

These groups can overlap. For example, the standard Whittaker functions can be written using the confluent hypergeometric function of the first kind and the confluent hypergeometric function of the second kind. The Meijer G-function is defined by a Mellin-Barnes integral and reduces to many other special functions for particular parameters.

See also

Elementary Function, Function

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References

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, 1972.Andrews, G. E.; Askey, R.; and Roy, R. Special Functions. Cambridge, England: Cambridge University Press, 1999.Arscott, F. M. "The Land beyond Bessel: A Survey of Higher Special Functions." In Ordinary and Partial Differential Equations (Ed. W. N. Everitt and B. D. Sleeman). New York: Springer-Verlag, pp. 26-45, 1981.Luke, Y. L. The Special Functions and Their Approximations, Vol. 1. New York: Academic Press, 1969.Luke, Y. L. The Special Functions and Their Approximations, Vol. 2. New York: Academic Press, 1969.Magnus, W. and Oberhettinger, F. Formulas and Theorems for the Special Functions of Mathematical Physics, 3rd ed. New York: Springer-Verlag, 1966.NIST Digital Library of Mathematical Functions. https://dlmf.nist.gov/.Nikiforov, A. F. and Uvarov, V. B. Special Functions of Mathematical Physics: A Unified Introduction with Applications. Boston, MA: Birkhäuser, 1988.Prudnikov, A. P.; Brychkov, Yu. A.; and Marichev, O. I. Integrals and Series, Vol. 1: Elementary Functions. New York: Gordon and Breach, 1986.Prudnikov, A. P.; Brychkov, Yu. A.; and Marichev, O. I. Integrals and Series, Vol. 3: More Special Functions. New York: Gordon and Breach, 1989.Prudnikov, A. P.; Brychkov, Yu. A.; and Marichev, O. I. Integrals and Series, Vol. 2: Special Functions. New York: Gordon and Breach, 1990.Prudnikov, A. P.; Brychkov, Yu. A.; and Marichev, O. I. Integrals and Series, Vol. 4: Direct Laplace Transforms. New York: Gordon and Breach, 1992.Prudnikov, A. P.; Brychkov, Yu. A.; and Marichev, O. I. Integrals and Series, Vol. 5: Inverse Laplace Transforms. New York: Gordon and Breach, 1992.Spanier, J.; Myland, J.; and Oldham, K. B. An Atlas of Functions, 2nd ed. Washington, DC: Hemisphere, 1987.Weisstein, E. W. "Books about Special Functions." http://www.ericweisstein.com/encyclopedias/books/SpecialFunctions.html.Wolfram Research, Inc. "Wolfram Research's Mathematical Functions." https://functions.wolfram.com/.

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Special Function

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Weisstein, Eric W. "Special Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SpecialFunction.html

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