Special functions


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 equationsor integrals of elementary functions. Therefore, tables of integrals cite book
last = Gradshtein
first = Israel Solomonovich
authorlink =
coauthors = Iosif Moiseevich Ryzhik.
title = Table of integrals, sums, series and products
publisher = Academic press
date =
location =
pages =
url =
doi =
id =
isbn =
] usually include description of special functions, and tables of special functions cite book
last = Abramovitz
first = Milton
authorlink =
coauthors = Irene Stegun
title = Table of mathematical functions
publisher =
date =
location =
pages =
url =
doi =
id =
isbn =
] include most important integrals; at least, the integral representation of special functions.

Languages for analytical calculus such as Mathematica [ [http://reference.wolfram.com/mathematica/guide/SpecialFunctions.html List of special functions in 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 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 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).

History of special functions

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.

pecial 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.

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


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • List of special functions and eponyms — This is a list of special function eponyms in mathematics, to cover the theory of special functions, the differential equations they satisfy, named differential operators of the theory (but not intended to include every mathematical eponym).… …   Wikipedia

  • Integral Transforms and Special Functions — is a scientific journal, specialised in topics of mathematical analysis, the theory of differential and integral equations, approximation theory, but publishes also papers in other areas of mathematics. It is published monthly by Taylor… …   Wikipedia

  • special function — ▪ mathematics       any of a class of mathematical functions (function) that arise in the solution of various classical problems of physics. These problems generally involve the flow of electromagnetic, acoustic, or thermal energy. Different… …   Universalium

  • special anatomy — study of particular organs; anatomy of specific definite organs or groups of organs and their special functions …   English contemporary dictionary

  • Special member functions — [cite book last = ISO/IEC authorlink = International Organization for Standardization title = International Standard ISO/IEC 14882: Programming languages C++ = Languages de programmation C++ edition = 1 publisher = ISO/IEC date = 1998 oclc =… …   Wikipedia

  • Special Devotions For Months —     Special Devotions for Months     † Catholic Encyclopedia ► Special Devotions for Months     During the Middle Ages the public functions of the Church and the popular devotions of the people were intimately connected. The laity assisted at the …   Catholic encyclopedia

  • Special Tasks and Rescue — (STAR Force) is the Police Tactical Group of the South Australia Police.HistoryFormed on 30 November, 1978 the South Australian Police STAR Force was a rationalisation of specialist resources into one command/unit. Specialist units had existed… …   Wikipedia

  • Special Organization (Ottoman Empire) — Special Organization was the name given to a three member executive committee of Ministry of the Interior established by the Committee of Union and Progress of the Ottoman Empire. It is speculated that the organization was planned and created… …   Wikipedia

  • Special European Union Programmes Body — Special EU Programs Body / Foras Um Chláir Speisialta An AE This is one of the six originally planned cross border/All Ireland/North South Implementation bodies set up following the Good Friday Agreement. Its functions are proscribed in law… …   Wikipedia

  • Special values of L-functions — In mathematics, the study of special values of L functions is a subfield of number theory devoted to generalising formulae such as the Leibniz formula for pi, namely by the recognition that expression on the left hand side is also L(1) where L(s) …   Wikipedia


Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.