Special functions

Special functions are particular mathematical functions which have more or less established names and notations due to their importance for the mathematical analysis, functional analysis, physics and 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.

Tables of special functions
Many special functions appear as solutions of differential equations or integrals of elementary functions. Therefore, tables of integrals usually include description of special functions, and tables of special functions include most important integrals; at least, the integral representation of special functions.

Languages for analytical calculus such as Mathematica 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 the most of cases, the standard notation is used for indication of a special function: the name of function (printed with Roman font), subscripts, if any, open parenthesis, and then arguments, separated with comma. Such a notation allows easy translation of the expressions to algorithmic languages avoiding ambiguities. Functions with established international notations are sin, cos, exp, erf, erfc.

Sometimes, a special function has several names. The natural logarithm can be called as Log, log or ln, dependently on the context. The tangent may be called as Tan, tan or tg (especially in Russian literature); arctangent can be called atan, arctg, $$\tan^{-1}$$. Function of Bessel can be called just $$~{\rm J}_n(x)~$$; usually, $$~J_n(x)~$$, $$~{\rm besselj}(n,x) ~$$, $$~ {\rm BesselJ}[n,x]~$$ refer to the same function.

Often the subscriptors are used to indicate argument(s), which is(are) usually supposed to be integer. In few faces, the semicolon or even backslash (\) is used as separator. Then, the translation to algorithmic languages allows ambiguity and may lead to confusions.

Superscript may indicate not only exponential, but modification of function. For example, $$~\cos^{3}(x)~$$, $$~\cos^{2}(x)~$$ $$~\cos^{-1}(x)~$$ may indicate $$~\cos(x)^3~$$, $$~\cos(x)^2~$$, $$~\cos(x)^{-1}~$$ (or $$~\arccos(x)~$$), respectively; but $$~\cos^2(x)~$$ almost never means $$~\cos(\cos(x))~$$.

Evaluation of special functions
Most of special functions are considered as a functions of complex variable(s). They are analytic; the singularities and cuts are described; the differential and integral representations are known and the expansion to the Taylor 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 simple 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, usually, the rational approximations are used, although, the rational approximations may be not so good in the case of complex argument(s).

Classical theory
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.

Changing and fixed motivations
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.

Twentieth century
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 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 at about the time when electronic computation was changing the motivations. Tabulation was no longer the main issue.

Contemporary theories
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 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.

Special functions in number theory
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.