# Special functions

In mathematics, there is a theory or theories of special functions, particular functions such as the trigonometric functions that have useful or attractive properties, and which occur in different applications often enough to warrant a name and attention of their own. There is no theory of general functions, as such; that is covered mainly by mathematical analysis and functional analysis.

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 the special function theory in the period 1850-1900 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 complex analysis techniques; 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 reasons for wanting it. 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:

• for numerical analysis, discovery of infinite series or other analytical expression allowing rapid calculation; and
• reduction of as many functions as possible to the given function.

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 textbook sought to unify the theory by using complex variables; the G. N. Watson tome 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 attempted to be encyclopedic, at about the time when electronic computation was changing the motivations, and tabulation no longer was the main issue. The 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 Lie group terms. Further, the 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.