# Special functions

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**Special functions** are particular [mathematical functions](/source/Function_(mathematics)) that have more or less established names and [notations](/source/Mathematical_notation) due to their importance in [mathematical analysis](/source/Mathematical_analysis), [functional analysis](/source/Functional_analysis), [geometry](/source/Geometry), [physics](/source/Physics), or other applications.

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

## Tables of special functions

Many special functions appear as solutions of [differential equations](/source/Differential_equation) or [integrals](/source/Integral) of [elementary functions](/source/Elementary_functions). Therefore, tables of integrals[1] usually include descriptions of special functions, and tables of special functions[2] 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 groups](/source/Lie_group) and [Lie algebras](/source/Lie_algebra), as well as certain topics in [mathematical physics](/source/Mathematical_physics).

[Symbolic computation](/source/Computer_algebra) engines usually recognize the majority of special functions.

### Notations used for special functions

Functions with established international notations are the [sine](/source/Sine) (\sin), [cosine](/source/Cosine) (\cos), [exponential function](/source/Exponential_function) (\exp), and [error function](/source/Error_function) (\operatorname{erf} or \operatorname{erfc}).

Some special functions have several notations:

- The [natural logarithm](/source/Natural_logarithm) may be denoted \ln, \log, \log_e, or \operatorname{Log} depending on the context.
- The [tangent](/source/Trigonometric_functions) function may be denoted \tan, \operatorname{Tan}, or \operatorname{tg} (used in several European languages).
- [Arctangent](/source/Arctangent) may be denoted \arctan, \operatorname{atan}, \operatorname{arctg}, or \tan^{-1}.
- The [Bessel functions](/source/Bessel_function) may be denoted - J_n(x), - \operatorname{besselj}(n,x), - {\rm BesselJ}[n,x].

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](/source/Trigonometric_function) and [hyperbolic functions](/source/Hyperbolic_function)) include:

- \cos^3(x) usually means (\cos(x))^3
- \cos^2(x) is typically (\cos(x))^2, but never \cos(\cos(x))
- \cos^{-1}(x) usually means \arccos(x), not (\cos(x))^{-1}; 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](/source/Complex_number) variable. They are [analytic](/source/Analytic_function); the singularities and cuts are described; the differential and integral representations are known and the expansion to the [Taylor series](/source/Taylor_series) or [asymptotic series](/source/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, [rational approximations](/source/Pade_approximation) are typically used, although they may behave badly in the case of complex argument(s).

## History of special functions

### Classical theory

While [trigonometry](/source/Trigonometry) and [exponential functions](/source/Exponential_function) 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 functions](/source/Elliptic_function); treatises that were essentially complete, such as that of [Tannery](/source/Jules_Tannery) and [Molk](/source/Jules_Molk),[3] expounded all the basic identities of the theory using techniques from [analytic function](/source/Analytic_function) theory (based on [complex analysis](/source/Complex_analysis)). The end of the century also saw a very detailed discussion of [spherical harmonics](/source/Spherical_harmonic).

### Changing and fixed motivations

While [pure mathematicians](/source/Pure_mathematics) 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](/source/Applied_mathematics). Applications to the physical sciences and engineering determined the relative importance of functions. Before [electronic computation](/source/Electronic_computer), the importance of a special function was affirmed by the laborious computation of extended [tables of values](/source/Mathematical_table) for ready [look-up](/source/Look-up_table), as for the familiar [logarithm tables](/source/Logarithm_tables). (Babbage's [difference engine](/source/Difference_engine) was an attempt to compute such tables.) For this purpose, the main techniques are:

- [numerical analysis](/source/Numerical_analysis), the discovery of [infinite series](/source/Infinite_series) or other [analytical expressions](/source/Analytical_expression) allowing rapid calculation; and
- reduction of as many functions as possible to the given function.

More theoretical questions include: [asymptotic analysis](/source/Asymptotic_analysis); [analytic continuation](/source/Analytic_continuation) and [monodromy](/source/Monodromy) in the [complex plane](/source/Complex_plane); and [symmetry](/source/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](/source/Whittaker_and_Watson)* (1902) textbook[4] sought to unify the theory using complex analysis; the [G. N. Watson](/source/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](/source/Bateman_Manuscript_Project), under the editorship of [Arthur Erdélyi](/source/Arthur_Erd%C3%A9lyi), 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](/source/Orthogonal_polynomials) is of a definite but limited scope. [Hypergeometric series](/source/Hypergeometric_series), observed by [Felix Klein](/source/Felix_Klein) to be important in [astronomy](/source/Astronomy) and [mathematical physics](/source/Mathematical_physics),[5] became an intricate theory, requiring later conceptual arrangement. [Lie group](/source/Lie_group) [representations](/source/Representation_theory) give an immediate generalization of [spherical functions](/source/Zonal_spherical_function); from 1950 onwards substantial parts of classical theory were recast in terms of Lie groups. Further, work on [algebraic combinatorics](/source/Algebraic_combinatorics) also revived interest in older parts of the theory. Conjectures of [Ian G. Macdonald](/source/Ian_G._Macdonald) helped open up large and active new fields with a special function flavour. [Difference equations](/source/Difference_equation) have begun to take their place beside [differential equations](/source/Differential_equation) as a source of special functions.

## Special functions in number theory

In [number theory](/source/Number_theory), certain special functions have traditionally been studied, such as particular [Dirichlet series](/source/Dirichlet_series) and [modular forms](/source/Modular_form). Almost all aspects of special function theory are reflected there, as well as some new ones, such as came out of [monstrous moonshine](/source/Monstrous_moonshine) theory.

## Special functions of matrix arguments

Analogues of several special functions have been defined on the space of [positive definite matrices](/source/Positive_definite_matrix), among them the power function which goes back to [Atle Selberg](/source/Atle_Selberg),[6] the [multivariate gamma function](/source/Multivariate_gamma_function),[7] and types of [Bessel functions](/source/Bessel_functions).[8]

The [NIST](/source/National_Institute_of_Standards_and_Technology) Digital Library of Mathematical Functions has a section covering several special functions of matrix arguments.[9]

## Researchers

- [George Andrews](/source/George_Andrews_(mathematician))
- [Richard Askey](/source/Richard_Askey)
- [Harold Exton](/source/Harold_Exton)
- [George Gasper](/source/George_Gasper)
- [Wolfgang Hahn](/source/Wolfgang_Hahn)
- [Mizan Rahman](/source/Mizan_Rahman)
- [Mourad E. H. Ismail](/source/Mourad_E._H._Ismail)
- [Tom Koornwinder](/source/Tom_Koornwinder)
- [Waleed Al-Salam](/source/Waleed_Al-Salam)
- [Dennis Stanton](/source/Dennis_Stanton)
- [Theodore S. Chihara](/source/Theodore_S._Chihara)
- [James A. Wilson](/source/James_A._Wilson)
- [Erik Koelink](/source/Erik_Koelink)
- [Eric Rains](/source/Eric_Rains)
- [Arpad Baricz](/source/Arpad_Baricz)

## See also

- [List of mathematical functions](/source/List_of_mathematical_functions)
- [List of special functions and eponyms](/source/List_of_special_functions_and_eponyms)
- [Elementary function](/source/Elementary_function)

## References

1. Zwillinger, Daniel & Moll, Victor Hugo (eds.) (2015 [October 2014]). *Table of Integrals, Series, and Products*. 8 ed. Translated by Scripta Technica, Inc. [Academic Press, Inc.](/source/Academic_Press,_Inc.). ISBN 978-0-12-384933-5. LCCN 2014010276.

1. Abramowitz, Milton & Stegun, Irene A. (1964). [*Handbook of Mathematical Functions*](https://archive.org/details/handbookofmathem1964abra). U.S. Department of Commerce, National Bureau of Standards.

1. Tannery, Jules (1972). [*Éléments de la théorie des fonctions elliptiques*](http://worldcat.org/oclc/310702720). Chelsea. ISBN 0-8284-0257-4. [OCLC 310702720](https://www.worldcat.org/oclc/310702720)

1. Whittaker, E. T. & Watson, G. N. (1996-09-13). [*A Course of Modern Analysis*](http://dx.doi.org/10.1017/cbo9780511608759). Cambridge University Press. [doi:10.1017/cbo9780511608759](https://doi.org/10.1017/cbo9780511608759). ISBN 978-0-521-58807-2.

1. Vilenkin, N.J. (1968). *Special Functions and the Theory of Group Representations*. Providence, RI: [American Mathematical Society](/source/American_Mathematical_Society). p. iii. ISBN 978-0-8218-1572-4.

1. Terras 2016, p. 44.

1. Terras 2016, p. 47.

1. Terras 2016, pp. 56ff.

1. [D. St. P. Richards](/source/Donald_Richards_(statistician)) (n.d.). ["Chapter 35 Functions of Matrix Argument"](https://dlmf.nist.gov/35). *[Digital Library of Mathematical Functions](/source/Digital_Library_of_Mathematical_Functions)*. Retrieved 23 July 2022.

### Bibliography

- Andrews, George E.; Askey, Richard; Roy, Ranjan (1999). *Special functions*. Vol. 71. Encyclopedia of Mathematics and its Applications. [Cambridge University Press](/source/Cambridge_University_Press). ISBN 978-0-521-62321-6. MR 1688958.
- Terras, Audrey (2016). *Harmonic analysis on symmetric spaces – Higher rank spaces, positive definite matrix space and generalizations*. second ed. [Springer Nature](/source/Springer_Nature). ISBN 978-1-4939-3406-5. MR 3496932.
- 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).
- Claude Brezinski and Michela Redivo-Zaglia: *The Birth and Early Developments of Orthogonal Polynomials: A Chronological History*, SIAM, ISBN 978-1-61197-850-6 (2025).

### 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).
- Nico M. Temme: ["Numerical aspects of special functions"](https://ir.cwi.nl/pub/11612/11612C.pdf). *[Acta Numerica](/source/Acta_Numerica)* (2008), pp. 1–101

## External links

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

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