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Difference between revisions of "Special functions"

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In the broad sense, a set of several classes of functions that arise in the solution of both theoretical and applied problems in various branches of mathematics.
 
In the broad sense, a set of several classes of functions that arise in the solution of both theoretical and applied problems in various branches of mathematics.
  
In the narrow sense, special functions are the special functions of mathematical physics, which arise when solving partial differential equations by the method of separation of variables (cf.
+
In the narrow sense, special functions are the special functions of mathematical physics, which arise when solving partial differential equations by the
[[Separation of variables, method of|Separation of variables, method of]]).
+
[[Separation of variables, method of|method of separation of variables]].
  
 
Special functions can be defined by means of power series, generating functions, infinite products, repeated differentiation, integral representations, differential, difference, integral, and functional equations, trigonometric series, or other series in orthogonal functions.
 
Special functions can be defined by means of power series, generating functions, infinite products, repeated differentiation, integral representations, differential, difference, integral, and functional equations, trigonometric series, or other series in orthogonal functions.
Line 11: Line 11:
 
The most important classes of special functions are the following: the
 
The most important classes of special functions are the following: the
 
[[Gamma-function|gamma-function]] and the
 
[[Gamma-function|gamma-function]] and the
[[Beta-function|beta-function]]; hypergeometric functions and confluent hypergeometric functions (cf.
+
[[Beta-function|beta-function]];  
[[Hypergeometric function|Hypergeometric function]];
+
[[Hypergeometric function|hypergeometric functions]] and
[[Confluent hypergeometric function|Confluent hypergeometric function]]);
+
[[Confluent hypergeometric function|confluent hypergeometric functions]];
[[Bessel functions|Bessel functions]];
+
[[Bessel functions]];
[[Legendre functions|Legendre functions]]; parabolic cylinder functions (cf.
+
[[Legendre functions]];  
[[Parabolic cylinder function|Parabolic cylinder function]]);
+
[[Parabolic cylinder function|parabolic cylinder functions]];
 
[[Integral sine|integral sine]] and
 
[[Integral sine|integral sine]] and
[[Integral cosine|integral cosine]] functions; incomplete gamma- and beta-functions (cf.
+
[[Integral cosine|integral cosine]] functions;
[[Incomplete gamma-function|Incomplete gamma-function]];
+
[[Incomplete gamma-function|incomplete gamma-]] and
[[Incomplete beta-function|Incomplete beta-function]]); probability integrals (cf.
+
[[Incomplete beta-function|beta-functions]];
[[Probability integral|Probability integral]]); various classes of
+
[[Probability integral|probability integrals]]; various classes of
[[Orthogonal polynomials|orthogonal polynomials]] in one or several variables; elliptic functions and integrals (cf.
+
[[Orthogonal polynomials|orthogonal polynomials]] in one or several variables;
[[Elliptic function|Elliptic function]];
+
[[Elliptic function|elliptic functions]] and
[[Elliptic integral|Elliptic integral]]); Lamé functions (cf.
+
[[Elliptic integral|integrals]];
[[Lamé function|Lamé function]]) and
+
[[Lamé function|Lamé functions]] and
[[Mathieu functions|Mathieu functions]]; the
+
[[Mathieu functions]]; the
[[Riemann zeta-function|Riemann zeta-function]]; automorphic functions (cf.
+
[[Riemann zeta-function]];
[[Automorphic function|Automorphic function]]); and some special functions of a discrete argument.
+
[[Automorphic function|automorphic functions]]; and some special functions of a discrete argument.
  
 
The theory of special functions is connected with the representation of groups (cf.
 
The theory of special functions is connected with the representation of groups (cf.
[[Representation theory|Representation theory]]), with methods of integral representations based on the generalization of the
+
[[Representation theory]]), with methods of integral representations based on the generalization of the
[[Rodrigues formula|Rodrigues formula]] for classical orthogonal polynomials, and with methods in
+
[[Rodrigues formula]] for classical orthogonal polynomials, and with methods in
 
[[Probability theory|probability theory]].
 
[[Probability theory|probability theory]].
  

Revision as of 12:52, 26 April 2012

2010 Mathematics Subject Classification: Primary: 33-XX [MSN][ZBL]

In the broad sense, a set of several classes of functions that arise in the solution of both theoretical and applied problems in various branches of mathematics.

In the narrow sense, special functions are the special functions of mathematical physics, which arise when solving partial differential equations by the method of separation of variables.

Special functions can be defined by means of power series, generating functions, infinite products, repeated differentiation, integral representations, differential, difference, integral, and functional equations, trigonometric series, or other series in orthogonal functions.

The most important classes of special functions are the following: the gamma-function and the beta-function; hypergeometric functions and confluent hypergeometric functions; Bessel functions; Legendre functions; parabolic cylinder functions; integral sine and integral cosine functions; incomplete gamma- and beta-functions; probability integrals; various classes of orthogonal polynomials in one or several variables; elliptic functions and integrals; Lamé functions and Mathieu functions; the Riemann zeta-function; automorphic functions; and some special functions of a discrete argument.

The theory of special functions is connected with the representation of groups (cf. Representation theory), with methods of integral representations based on the generalization of the Rodrigues formula for classical orthogonal polynomials, and with methods in probability theory.

There are tables of values of special functions and also tables of integrals and series.

Comments

Given a Lie group $G$ and a (matrix) representation $\rho$ of it, one can regard the matrix coefficients of $\rho$ as functions on $G$. Many special functions can be seen as arising essentially in this way, and this point of view "explains" many of the special properties of special functions, e.g. various orthogonality relations. Cf. [Vi], [Mi], [Wa2], and the encyclopaedic treatment [ViKl], vol. 1, for more details.

Many special functions have so-called $q$-analogues, $q$-special functions. That means, roughly, that it is possible to insert a parameter $q$ to obtain a family of special functions in such a way that many of the characteristic properties of special functions are retained. These $q$-special functions correspond to quantum groups in the same way that special functions relate to Lie groups. Cf. the recent survey [Ko], and [ViKl], vols. 2–3, for more details.


References

[AbSt] M. Abramowitz, I.A. Stegun, "Handbook of mathematical functions", Dover, reprint (1970) MR0415962 Zbl 0643.33002 Zbl 0643.33001 Zbl 0515.33001 Zbl 0543.33001 Zbl 0171.38503
[Er] A. Erdélyi et al. (ed.), Higher transcendental functions, 1–3, McGraw-Hill (1953–1955) MR0058756 Zbl 0052.29502 Zbl 0051.30303
[Fe] W. Feller, "An introduction to probability theory and its applications", 1–2, Wiley (1957–1971) MR0088081 Zbl 0077.12201
[GrRy] I.S. Gradshtein, I.M. Ryzhik, "Table of integrals, series and products", Acad. Press (1980) (Translated from Russian) MR0582453
[Ho] E.W. Hobson, "The theory of spherical and ellipsoidal harmonics", Cambridge Univ. Press (1931) MR0064922 MR1522948 Zbl 0004.21001 JFM Zbl 57.0405.06
[JaEmLö] E. Jahnke, F. Emde, F. Lösch, "Tafeln höheren Funktionen", Teubner (1966)
[Ko] T.H. Koornwinder, "Orthogonal polynomials in connection with quantum groups" P. Nevai (ed.), Orthogonal polynomials: theory and practice, Kluwer (1990) pp. 257–292 MR2281172
[KrFr] A. Krazer, W. Franz, "Transzendente Funktionen", Akademie Verlag (1960)
[Le] N.N. Lebedev, "Special functions and their applications", Prentice-Hall (1965) (Translated from Russian) MR0174795 Zbl 0131.07002
[Mi] W. Miller jr., "Lie theory and special functions", Acad. Press (1968) MR0264140 Zbl 0174.10502
[NiUf] A.F. Nikiforov, V.B. Ufarov, "Special functions of mathematical physics", Birkhäuser (1988) (Translated from Russian) MR0922041 Zbl 0624.33001
[PrBrMa] A.P. Prudnikov, Yu.A. Brychkov, O.I. Marichev, "Integrals and series. Elementary functions", Gordon & Breach (1986) (Translated from Russian) MR0888165 MR0874987 MR0874986 Zbl 0511.00044
[PrBrMa2] A.P. Prudnikov, Yu.A. Brychkov, O.I. Marichev, "Integrals and series. Special functions", Gordon & Breach (1986) (Translated from Russian) MR0888165 MR0874987 MR0874986 Zbl 0626.00033 Zbl 0733.00004
[PrBrMa3] A.P. Prudnikov, Yu.A. Brychkov, O.I. Marichev, "Integrals and series. Additional chapters", Gordon & Breach (1987) (Translated from Russian) MR1162980 MR1162979 MR1054647
[SrKa] H.M. Srivastava, B.R.K. Kashyap, "Special functions in queuing theory", Acad. Press (1982) MR0657766 Zbl 0492.60089
[Sz] G. Szegö, "Orthogonal polynomials", Amer. Math. Soc. (1975) MR0310533
[Vi] N.Ya. Vilenkin, "Special functions and the theory of group representations", Amer. Math. Soc. (1968) (Translated from Russian) MR0229863 Zbl 0172.18404
[ViKl] N.Ya. Vilenkin, A.U. Klimyk, "Representations of Lie groups, special functions and integral transforms", 1–3, Kluwer (1991-) (Translated from Russian)
[Wa] G.N. Watson, "A treatise on the theory of Bessel functions", 1–2, Cambridge Univ. Press (1952) MR1349110 Zbl 0849.33001
[Wa2] A. Wawrzyńczyk, "Group representations and special functions", Reidel (1984) MR0750113 Zbl 0545.43001
[WhWa] E.T. Whittaker, G.N. Watson, "A course of modern analysis", Cambridge Univ. Press (1952) pp. Chapt. 6 MR1424469 Zbl 0951.30002
How to Cite This Entry:
Special functions. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Special_functions&oldid=25500
This article was adapted from an original article by Yu.A. BrychkovA.P. Prudnikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article