- Special functions
**Special functions**are particular mathematical functions which have more or less established names and notations due to their importance for themathematical 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 ofdifferential equation sorintegral s ofelementary functions . Therefore, tables of integrals cite book

last = Gradshtein

first = Israel Solomonovich

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coauthors = Iosif Moiseevich Ryzhik.

title = Table of integrals, sums, series and products

publisher = Academic press

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last = Abramovitz

first = Milton

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coauthors = Irene Stegun

title = Table of mathematical functions

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isbn = ] 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.*http://reference.wolfram.com/mathematica/guide/SpecialFunctions.html List of special functions in Mathematica*]**Notations used in special functions**In most 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, depending on the context. The tangent may be called as Tan, tan or tg (especially in Russian literature); arctangent can be called atan, arctg, $an^\{-1\}$. Function of Bessel can be called just$~\{\; m\; J\}\_n(x)~$; usually, $~J\_n(x)~$,$~\{\; m\; besselj\}(n,x)\; ~$, $~\; \{\; m\; BesselJ\}\; [n,x]\; ~$ 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 orasymptotic 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 aTaylor 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).**History of special functions****Classical theory**While

trigonometry can be codified, as was clear already to expert mathematicians of theeighteenth century (if not before), the search for a complete and unified theory of special functions has continued since thenineteenth century . The high point of the special function theory in the period 1850-1900 was the theory ofelliptic function s; treatises that were essentially complete, such as that ofTannery and Molk , could be written as handbooks to all the basic identities of the theory. They were based oncomplex analysis techniques.From that time onwards it would be assumed that

analytic function theory, which had already unified the trigonometric andexponential function s, was a fundamental tool. The end of the century also saw a very detailed discussion ofspherical harmonic s.**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 theelectronic 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 bylook-up , as for the familiarlogarithm tables . The aspects of the theory that then mattered might then be two:*for

numerical analysis , discovery ofinfinite series or otheranalytical 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 andmonodromy in thecomplex plane , and the discovery ofsymmetry 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 usingcomplex variable s; theG. 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 ofArthur 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 polynomial s is of a definite but limited scope.Hypergeometric series became an intricate theory, in need of later conceptual arrangement.Lie group s, and in particular theirrepresentation theory , explain what aspherical function can be in general; from 1950 onwards substantial parts of classical theory could be recast in Lie group terms. Further, the work onalgebraic combinatorics also revived interest in older parts of the theory. Conjectures ofIan G. Macdonald helped to open up large and active new fields with the typical special function flavour.Difference equation s have begun to take their place besidesdifferential equation s as a source for special functions.**pecial functions in number theory**In

number theory certain special functions have traditionally been studied, such as particularDirichlet series andmodular form s. Almost all aspects of special function theory are reflected there, as well as some new ones, such as came out of themonstrous moonshine theory.**ee also***

List of mathematical functions

*List of special functions and eponyms **References***Citation | last1=Andrews | first1=George E. | last2=Askey | first2=Richard | last3=Roy | first3=Ranjan | title=Special functions | publisher=

Cambridge University Press | series=Encyclopedia of Mathematics and its Applications | isbn=978-0-521-62321-6; 978-0-521-78988-2 | id=MathSciNet | id = 1688958 | year=1999 | volume=71**External links*** [

*http://www.special-functions.com Special functions*] , Special functions calculator.

* [*http://eqworld.ipmnet.ru/en/auxiliary/aux-specfunc.htm Special functions*] at EqWorld: The World of Mathematical Equations.

*Abramowitz and Stegun , handbook on special functions.

* http://www.amazon.ca/exec/obidos/ASIN/0122947576, Gradshtein, Ryshik

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