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.
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.
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. In this case, the translation to algorithmic languages admits ambiguity and may lead to confusion.
Most special functions are considered as a function of a complex variable. They are analytic; the singularities and cuts are described; the differential and integral representations are known and the expansion to the Taylor series or asymptotic 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 if at all. In algorithmic languages, rational approximations are typically used, although they may behave badly in the case of complex argument(s).
While trigonometry can be codified – as was clear already to expert mathematicians of the eighteenth century (if not before) – 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 the period 1800-1900 was the theory of elliptic functions; treatises that were essentially complete, such as that of Tannery and Molk, could be written as handbooks to all the basic identities of the theory. They were based on techniques from complex analysis.
From that time onwards it would be assumed that analytic function theory, which had already unified the trigonometric and exponential functions, was a fundamental tool. The end of the century also saw a very detailed discussion of spherical harmonics.
Of course the wish for a broad theory including as many as possible of the known special functions has its intellectual appeal, but it is worth noting other motivations. For a long time, the special functions were in the particular province of applied mathematics; applications to the physical sciences and engineering determined the relative importance of functions. In the days before the electronic computer, the ultimate compliment to a special function was the computation, by hand, of extended tables of its values. This was a capital-intensive process, intended to make the function available by look-up, as for the familiar logarithm tables. The aspects of the theory that then mattered might then be two:
In contrast, one might say, there are approaches typical of the interests of pure mathematics: asymptotic analysis, analytic continuation and monodromy in the complex plane, and the discovery of symmetry principles and other structure behind the façade of endless formulae in rows. There is not a real conflict between these approaches, in fact.
The twentieth century saw several waves of interest in special function theory. The classic Whittaker and Watson (1902) textbook sought to unify the theory by using complex variables; the G. N. Watson tome A Treatise on the Theory of Bessel Functions pushed the techniques as far as possible for one important type that particularly admitted asymptotics to be studied.
The later Bateman Manuscript Project, under the editorship of Arthur 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 of orthogonal polynomials is of a definite but limited scope. Hypergeometric series became an intricate theory, in need of later conceptual arrangement. Lie groups, and in particular their representation theory, explain what a spherical function can be in general; from 1950 onwards substantial parts of classical theory could be recast in terms of Lie groups. Further, work on algebraic combinatorics also revived interest in older parts of the theory. Conjectures of Ian G. Macdonald helped to open up large and active new fields with the typical special function flavour. Difference equations have begun to take their place besides differential equations as a source for special functions.
In number theory, certain special functions have traditionally been studied, such as particular Dirichlet series and modular forms. Almost all aspects of special function theory are reflected there, as well as some new ones, such as came out of the monstrous moonshine theory.