Special functions

Summary

Special functions are the named functions of analysis that occur again and again in mathematics and physics, beyond the elementary functions. Examples are the gamma function, the hypergeometric function, the Bessel functions, the Legendre polynomials and the elliptic integrals. The term has no formal definition: a function counts as special when it has an established name, notation and theory. Their study collects identities, integral representations, series, asymptotic formulas and differential equations that make these functions usable in computation.