Special functions

Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications. The term is defined by consensus, and thus lacks a general formal definition, but the list of mathematical functions contains functions that are commonly accepted as special. Many special functions appear as solutions of differential equations or integrals of elementary functions. Therefore, tables of integrals usually include descriptions of special functions, and tables of special functions 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. Functions with established international notations are the sine (), cosine (), exponential function (), and error function ( or ). Some special functions have several notations: The natural logarithm may be denoted , , , or depending on the context. The tangent function may be denoted , , or ( is used in several European languages). Arctangent may be denoted , , , or . The Bessel functions may be denoted Subscripts are often used to indicate arguments, typically integers. In a few cases, the semicolon (;) or even backslash () is used as a separator. In this case, the translation to algorithmic languages admits ambiguity and may lead to confusion. Superscripts may indicate not only exponentiation, but modification of a function. Examples (particularly with trigonometric and hyperbolic functions) include: usually means is typically , but never usually means , not ; this one typically causes the most confusion, since the meaning of this superscript is inconsistent with the others.