View source for Special functions

{{short description|Mathematical functions having established names and notations}}
{{more citations needed|date=July 2021}}
'''Special functions''' are particular [[function (mathematics)|mathematical function]]s that have more or less established names and [[mathematical notation|notation]]s due to their importance in [[mathematical analysis]], [[functional analysis]], [[geometry]], [[physics]], or other applications.

The term is defined by consensus, and thus lacks a general formal definition, but the [[list of mathematical functions]] contains functions that are commonly accepted as special.

==Tables of special functions==
Many special functions appear as solutions of [[differential equation]]s or [[integral]]s of [[elementary functions]]. Therefore, tables of integrals<ref name="Zwillinger_2014">{{cite book |author-first1=Izrail Solomonovich |author-last1=Gradshteyn |author-link1=Izrail Solomonovich Gradshteyn |author-first2=Iosif Moiseevich |author-last2=Ryzhik |author-link2=Iosif Moiseevich Ryzhik |author-first3=Yuri Veniaminovich |author-last3=Geronimus |author-link3=Yuri Veniaminovich Geronimus |author-first4=Michail Yulyevich |author-last4=Tseytlin |author-link4=Michail Yulyevich Tseytlin |author-first5=Alan |author-last5=Jeffrey |editor-first1=Daniel |editor-last1=Zwillinger |editor-first2=Victor Hugo |editor-last2=Moll |editor-link2=Victor Hugo Moll |translator=Scripta Technica, Inc. |title=Table of Integrals, Series, and Products |publisher=[[Academic Press, Inc.]] |date=2015 |orig-year=October 2014 |edition=8 |language=en |isbn=978-0-12-384933-5 |lccn=2014010276 <!-- |url=https://books.google.com/books?id=NjnLAwAAQBAJ |access-date=2016-02-21-->|title-link=Gradshteyn and Ryzhik}}</ref> usually include descriptions of special functions, and tables of special functions<ref name="IRENE">
{{cite book
  | last1 = Abramowitz
  | first1 = Milton
  | author-link1 = Milton Abramowitz
  | first2 = Irene A. |last2 = Stegun |author-link2 = Irene Stegun 
  | title = Handbook of Mathematical Functions 
  | date = 1964
  | publisher = U.S. Department of Commerce, National Bureau of Standards
 | url = https://archive.org/details/handbookofmathem1964abra
}}</ref> 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 group]]s and [[Lie algebra]]s, as well as certain topics in [[mathematical physics]].

[[Computer algebra|Symbolic computation]] engines usually recognize the majority of special functions.

===Notations used for special functions===
Functions with established international notations are the [[sine]] (<math>\sin</math>), [[cosine]] (<math>\cos</math>), [[exponential function]] (<math>\exp</math>), and [[error function]] (<math>\operatorname{erf}</math> or <math>\operatorname{erfc}</math>).

Some special functions have several notations:

* The [[natural logarithm]] may be denoted <math>\ln</math>, <math>\log</math>, <math>\log_e</math>, or <math>\operatorname{Log}</math> depending on the context. 
* The [[Trigonometric functions#tangent|tangent]]{{Broken anchor|date=2024-09-29|bot=User:Cewbot/log/20201008/configuration|target_link=Trigonometric functions#tangent|reason= The anchor (tangent) [[Special:Diff/952491828|has been deleted]].}} function may be denoted <math>\tan</math>, <math>\operatorname{Tan}</math>, or <math>\operatorname{tg}</math> (used in several European languages).
* [[Arctangent]] may be denoted <math>\arctan</math>, <math>\operatorname{atan}</math>, <math>\operatorname{arctg}</math>, or <math>\tan^{-1}</math>.
* The [[Bessel function]]s may be denoted
** <math>J_n(x),</math>
** <math>\operatorname{besselj}(n,x),</math>
** <math>{\rm BesselJ}[n,x].</math>

Subscripts are often used to indicate arguments, typically integers. In a few cases, the semicolon (;) or even backslash (\) is used as a separator for arguments. This may confuse the translation to algorithmic languages.

Superscripts may indicate not only a power (exponent), but some other modification of the function. Examples (particularly with [[trigonometric function|trigonometric]] and [[hyperbolic function]]s) include:

* <math>\cos^3(x)</math> usually means <math>(\cos(x))^3</math>
* <math>\cos^2(x)</math> is typically <math>(\cos(x))^2</math>, but never <math>\cos(\cos(x))</math>
* <math>\cos^{-1}(x)</math> usually means <math>\arccos(x)</math>, not <math>(\cos(x))^{-1}</math>; this may cause confusion, since the meaning of this superscript is inconsistent with the others.

===Evaluation of special functions===
Most special functions are considered as a function of a [[complex number|complex]] variable. They are [[analytic function|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 or not at all. In algorithmic languages, [[Pade approximation|rational approximation]]s are typically used, although they may behave badly in the case of complex argument(s).

==History of special functions==

===Classical theory===
While [[trigonometry]] and [[exponential function]]s were systematized and unified by the eighteenth century, 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 1800–1900 was the theory of [[elliptic function]]s; treatises that were essentially complete, such as that of [[Jules Tannery|Tannery]] and [[Jules Molk|Molk]],<ref>{{Cite book |first=Jules |last=Tannery |url=http://worldcat.org/oclc/310702720 |title=Éléments de la théorie des fonctions elliptiques |date=1972 |publisher=Chelsea |isbn=0-8284-0257-4 |oclc=310702720}}</ref> expounded all the basic identities of the theory using techniques from [[analytic function]] theory (based on [[complex analysis]]). The end of the century also saw a very detailed discussion of [[spherical harmonic]]s.

===Changing and fixed motivations===
While [[Pure mathematics|pure mathematicians]] sought a broad theory deriving as many as possible of the known special functions from a single principle, for a long time the special functions were the province of [[applied mathematics]]. Applications to the physical sciences and engineering determined the relative importance of functions. Before [[electronic computer|electronic computation]], the importance of a special function was affirmed by the laborious computation of extended [[Mathematical table|tables of values]] for ready [[Look-up table|look-up]], as for the familiar [[logarithm tables]]. (Babbage's [[difference engine]] was an attempt to compute such tables.) For this purpose, the main techniques are:

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

More theoretical questions include: [[asymptotic analysis]]; [[analytic continuation]] and [[monodromy]] in the [[complex plane]]; and [[symmetry]] principles and other structural equations.

===Twentieth century===
The twentieth century saw several waves of interest in special function theory. The classic ''[[Whittaker and Watson]]'' (1902) textbook<ref>{{Cite book |last1=Whittaker |first1=E. T. |url=http://dx.doi.org/10.1017/cbo9780511608759 |title=A Course of Modern Analysis |last2=Watson |first2=G. N. |date=1996-09-13 |publisher=Cambridge University Press |doi=10.1017/cbo9780511608759 |isbn=978-0-521-58807-2}}</ref> sought to unify the theory using complex analysis; the [[G. N. Watson]] tome ''A Treatise on the Theory of Bessel Functions'' pushed the techniques as far as possible for one important type, including asymptotic results.

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.

===Contemporary theories===
The modern theory of [[orthogonal polynomials]] is of a definite but limited scope. [[Hypergeometric series]], observed by [[Felix Klein]] to be important in [[astronomy]] and [[mathematical physics]],<ref>{{cite book |last=Vilenkin |first=N.J.|author-link=Naum Ya. Vilenkin |date=1968 |title=Special Functions and the Theory of Group Representations |url= |location=Providence, RI |publisher=[[American Mathematical Society]] |page=iii |isbn=978-0-8218-1572-4}}</ref> became an intricate theory, requiring later conceptual arrangement. [[Lie group]] [[representation theory|representations]] give an immediate generalization of [[Zonal spherical function|spherical functions]]; from 1950 onwards substantial parts of classical theory were 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 open up large and active new fields with a special function flavour. [[Difference equation]]s have begun to take their place beside [[differential equation]]s as a source of special functions.

==Special functions in number theory==
In [[number theory]], certain special functions have traditionally been studied, such as particular [[Dirichlet series]] and [[modular form]]s. Almost all aspects of special function theory are reflected there, as well as some new ones, such as came out of [[monstrous moonshine]] theory.

==Special functions of matrix arguments==
Analogues of several special functions have been defined on the space of [[positive definite matrix|positive definite matrices]], among them the power function which goes back to [[Atle Selberg]],{{sfn|Terras|2016|p=44}} the [[multivariate gamma function]],{{sfn|Terras|2016|p=47}} and types of [[Bessel functions]].{{sfn|Terras|2016|pp=56ff}}

The [[National Institute of Standards and Technology|NIST]] Digital Library of Mathematical Functions has a section covering several special functions of matrix arguments.<ref>{{cite web|url=https://dlmf.nist.gov/35|title=Chapter 35 Functions of Matrix Argument|work=[[Digital Library of Mathematical Functions]]|author=[[Donald Richards (statistician)|D. St. P. Richards]]|date=n.d.|access-date=23 July 2022}}</ref>

==Researchers==
{{cols|colwidth=21em}}
* [[George Andrews (mathematician)|George Andrews]]
* [[Richard Askey]]
* [[Harold Exton]]
* [[George Gasper]]
* [[Wolfgang Hahn]]
* [[Mizan Rahman]]
* [[Mourad E. H. Ismail]]
* [[Tom Koornwinder]]
* [[Waleed Al-Salam]]
* [[Dennis Stanton]]
* [[Theodore S. Chihara]]
* [[James A. Wilson]]
* [[Erik Koelink]]
* [[Eric Rains]]
* [[Arpad Baricz]]
{{colend}}

==See also==
* [[List of mathematical functions]]
* [[List of special functions and eponyms]]
* [[Elementary function]]

==References==
<references/>

===Bibliography===
* {{Cite book | last1=Andrews | first1=George E. | author-link=George E. Andrews | last2=Askey | first2=Richard |author-link2=Richard Askey |  last3=Roy | first3=Ranjan | title=Special functions | publisher=[[Cambridge University Press]] | series=Encyclopedia of Mathematics and its Applications |isbn=978-0-521-62321-6 |mr=1688958 | year=1999 | volume=71}}
* {{Cite book | last=Terras | first=Audrey | author-link=Audrey Terras | title=Harmonic analysis on symmetric spaces{{snd}}Higher rank spaces, positive definite matrix space and generalizations| edition = second| publisher=[[Springer Nature]]  |isbn=978-1-4939-3406-5 |mr=3496932 | year=2016}}
* Whittaker, E. T.; Watson, G. N. (1996-09-13). ''A Course of Modern Analysis''. Cambridge University Press. {{ISBN|978-0-521-58807-2}}.
* N. N. Levedev (Translated & Edited by Richard A. Sliverman): ''Special Functions & Their Applications'', DOVER, ISBN 978-0-486-60624-8 (1972). # Originally published from Prentice-Hall Inc.(1965).
* Nico M. Temme: ''Special Functions: An Introduction to the Classical Functions of Mathematical Physics'', Wiley-Interscience,ISBN 978-0-471-11313-1 (1996).
* Yury A. Brychkov: ''Handbook of Special Functions: Derivatives, Integrals, Series and Other Formulas'', CRC Press, ISBN 978-1-58488-956-4 (2008).
* W. W. Bell: ''Special Functions : for Scientists and Engineers'', Dover, ISBN 978-0-486-43521-3 (2004).

=== Numerical calculation method of function value ===
* Shanjie Zhang and Jian-Ming Jin: ''Computation of Special Functions'', Wiley-Interscience, ISBN 978-0-471-11963-0 (1996).
* William J. Thompson: ''Atlas for Computing Mathematical Functions: An Illustrated Guide for Practitioners; With Programs in C and Mathematica'', Wiley-Interscience, ISBN 978-0-471-00260-4 (March, 1997).
* William J. Thompson: ''Atlas for Computing Mathematical Functions: An illustrated Guide for Practitioners; With Programs in Fortran 90 and Mathematica'', Wiley-Interscience, ISBN 978-0-471-18171-2 (June, 1997).
* Amparo Gil, Javier Segura and Nico M. Temme: ''Numerical Methods for Special Functions'', SIAM, ISBN 978-0-898716-34-4 (2007).

==External links==
* [[National Institute of Standards and Technology]], United States Department of Commerce. [https://dlmf.nist.gov ''NIST Digital Library of Mathematical Functions'']. [https://web.archive.org/web/20181213070412/https://dlmf.nist.gov/ Archived] from the original on December 13, 2018.
* {{MathWorld|id=SpecialFunction |title=Special Function}}
* [http://www.lamprechts.de/gerd/php/RechnerMitUmkehrfunktion.php Online calculator], Online scientific calculator with over 100 functions (>=32 digits, many complex) (German language)
* [http://eqworld.ipmnet.ru/en/auxiliary/aux-specfunc.htm Special functions] at ''EqWorld: The World of Mathematical Equations''
* [http://www.phys.uu.nl/~thooft/lectures/specialfct.pdf ''Special functions and polynomials''] by Gerard 't Hooft and Stefan Nobbenhuis (April&nbsp;8, 2013)
* [http://personales.unican.es/segurajj/book/ Numerical Methods for Special Functions], by A. Gil, J. Segura, N.M. Temme (2007).
* R. Jagannathan, [https://arxiv.org/abs/math/9803142 (P,Q)-Special Functions]
* [http://specialfunctionswiki.org Specialfunctionswiki]

{{Analysis-footer}}
{{Authority control}}

[[Category:Special functions| ]]
[[Category:History of mathematics]]