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Special functions are particularmathematical functions that have more or less established names andnotations due to their importance inmathematical analysis,functional analysis,geometry,physics, or other applications.
The term is defined by consensus, and thus lacks a general formal definition, but thelist of mathematical functions contains functions that are commonly accepted as special.
Many special functions appear as solutions ofdifferential equations orintegrals ofelementary 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 ofLie groups andLie algebras, as well as certain topics inmathematical physics.
Symbolic computation engines usually recognize the majority of special functions.
Functions with established international notations are thesine (),cosine (),exponential function (), anderror function ( or).
Some special functions have several notations:
Subscripts are often used to indicate arguments, typically integers. In a few cases, the semicolon (;) or even backslash (\) is used as a separator for arguments. This may confuse the translation to algorithmic languages.
Superscripts may indicate not only a power (exponent), but some other modification of the function. Examples (particularly withtrigonometric andhyperbolic functions) include:
Most special functions are considered as a function of acomplex variable. They areanalytic; the singularities and cuts are described; the differential and integral representations are known and the expansion to theTaylor series orasymptotic series are available. In addition, sometimes there exist relations with other special functions; a complicated special function can be expressed in terms of simpler functions. Various representations can be used for the evaluation; the simplest way to evaluate a function is to expand it into a Taylor series. However, such representation may converge slowly or not at all. In algorithmic languages,rational approximations are typically used, although they may behave badly in the case of complex argument(s).
Whiletrigonometry andexponential functions were systematized and unified by the eighteenth century, the search for a complete and unified theory of special functions has continued since the nineteenth century. The high point of special function theory in 1800–1900 was the theory ofelliptic functions; treatises that were essentially complete, such as that ofTannery andMolk,[3] expounded all the basic identities of the theory using techniques fromanalytic function theory (based oncomplex analysis). The end of the century also saw a very detailed discussion ofspherical harmonics.
Whilepure mathematicians sought a broad theory deriving as many as possible of the known special functions from a single principle, for a long time the special functions were the province ofapplied mathematics. Applications to the physical sciences and engineering determined the relative importance of functions. Beforeelectronic computation, the importance of a special function was affirmed by the laborious computation of extendedtables of values for readylook-up, as for the familiarlogarithm tables. (Babbage'sdifference engine was an attempt to compute such tables.) For this purpose, the main techniques are:
More theoretical questions include:asymptotic analysis;analytic continuation andmonodromy in thecomplex plane; andsymmetry principles and other structural equations.
The twentieth century saw several waves of interest in special function theory. The classicWhittaker and Watson (1902) textbook[4] sought to unify the theory using complex analysis; theG. N. Watson tomeA Treatise on the Theory of Bessel Functions pushed the techniques as far as possible for one important type, including asymptotic results.
The laterBateman Manuscript Project, under the editorship ofArthur Erdélyi, attempted to be encyclopedic, and came around the time when electronic computation was coming to the fore and tabulation ceased to be the main issue.
The modern theory oforthogonal polynomials is of a definite but limited scope.Hypergeometric series, observed byFelix Klein to be important inastronomy andmathematical physics,[5] became an intricate theory, requiring later conceptual arrangement.Lie grouprepresentations give an immediate generalization ofspherical functions; from 1950 onwards substantial parts of classical theory were recast in terms of Lie groups. Further, work onalgebraic combinatorics also revived interest in older parts of the theory. Conjectures ofIan G. Macdonald helped open up large and active new fields with a special function flavour.Difference equations have begun to take their place besidedifferential equations as a source of special functions.
Innumber theory, certain special functions have traditionally been studied, such as particularDirichlet series andmodular forms. Almost all aspects of special function theory are reflected there, as well as some new ones, such as came out ofmonstrous moonshine theory.
Analogues of several special functions have been defined on the space ofpositive definite matrices, among them the power function which goes back toAtle Selberg,[6] themultivariate gamma function,[7] and types ofBessel functions.[8]
TheNIST Digital Library of Mathematical Functions has a section covering several special functions of matrix arguments.[9]
