Special functions are particular mathematical functions which have more or less established names and notations due to their importance in mathematical analysis, functional analysis, physics, or other applications.
There is no general formal definition, but the list of mathematical functions contains functions which are commonly accepted as special. In particular, elementary functions are also considered as special functions.
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Tables of special functions
Many special functions appear as solutions of differential equations or integrals of elementary functions. Therefore, tables of integrals ^{[1]} usually include descriptions of special functions, and tables of special functions ^{[2]} include most important integrals; at least, the integral representation of special functions. Because symmetries of differential equations are essential to both physics and mathematics, the theory of special functions is closely related to the theory of Lie groups and Lie algebras, as well as certain topics in mathematical physics.
Symbolic computation engines usually recognize the majority of special functions. Not all such systems have efficient algorithms for the evaluation, especially in the complex plane.
Notations used in special functions
In most cases, the standard notation is used for indication of a special function: the name of function, subscripts, if any, open parenthesis, then arguments, separated with comma, and then close parenthesis. Such a notation allows easy translation of the expressions to algorithmic languages avoiding ambiguities. Functions with established international notations are sin, cos, exp, erf, and erfc.
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