Special function

In analysis , a branch of mathematics , certain functions are called special functions . These play a central role both in pure mathematics itself and in its applications, such as in mathematical physics . The concept of the special function is not precisely defined . Often, however, a special function is only understood to mean functions that are at least transcendent . Therefore, instead of a special function, one also speaks of a higher transcendent function .

Definition

The concept of the special function is not precisely defined. From a pragmatic point of view, a special function is usually a function that depends on a variable that is also not an elementary function such as the algebraic functions , the trigonometric functions , the exponential function , the logarithmic function and functions that are derived from these by means of algebraic operations can be constructed, and which is of such importance for mathematics or its applications that it is or has been the subject of intensive research and has been dealt with intensively in the relevant specialist literature.

Many special functions are among the transcendent functions and are also referred to as higher transcendent functions. A large part of the special functions are holomorphic or meromorphic and can therefore be developed in series .

Special functions are often closely related to one another. Therefore, it is the subject of research today to classify the special functions themselves and the relationships to one another. Since the 19th century , various approaches have been developed with which important special functions can be treated as special cases of groups of functions that can be represented in a closed manner . These include the Meijer G function , the Fox H function and the hypergeometric function .

List of some special functions

• The following special functions are generalizations of the faculty or the gamma function
  • Gamma function
  • Euler's beta function
  • Pochhammer Icon
  • Polygamma functions (special cases : digamma function , trigamma function )
  • Barnesian G function
• Hypergeometric functions
  • Special functions that result from the hypergeometric function for special parameters
    • Legendre polynomials
    • Hermitian polynomials
    • Laguerre polynomials
    • Bessel functions
• Functions as a generalized logarithms are used
  • Polylogarithms
  • Nielsen function
  • Clausen functions
• Zeta functions
  • Riemann zeta function and Riemann xi function
• Other special functions
  • Elliptic integral functions
  • Dirichlet eta function
  • Dirichlet's lambda function
  • Dirichlet's beta function
  • Airy function

In multi-dimensional analysis, special functions in several (usually complex) variables are also used.

• Special functions in several parameters
  • Generalized hypergeometric functions
    • Appell functions

Other special functions of theoretical physics :

• Clebsch Gordan symbols
• Wigner-nj symbols

application areas

Many of these functions are solutions of differential equations that occur in important application situations. Special functions are also the backbone of many calculations with computer algebra systems ( Mathematica , Maple , etc.).

More recently, the properties of special functions have also been studied using computer algebra and symbolic mathematics . They are of particular importance in analytical number theory .

literature

• Irene A. Stegun and Milton Abramowitz : Handbook of Mathematical Functions , Dover Press ( online version ).
• Arthur Erdélyi , Hans Heinrich Wilhelm Magnus , Fritz Oberhettinger and Francesco Tricomi : Higher Transcendental Functions , McGraw-Hill, New York 1953–1955, OCLC 627135 ( digitized versionhttp: //vorlage_digitalisat.test/1%3D~GB%3D~IA%3Dhighertranscende02bate%2Fpage%2Fn5~MDZ%3D%0A~SZ%3D~doppelseiten%3D~LT%3D~PUR%3D )

Web links

• NIST Digital Library of Mathematical Functions - National Institute of Standards and Technology (English)

Individual evidence

1. ^ R. Beals, R. Wong: Special functions. A graduate text . Cambridge University Press, New York 2010, ISBN 978-0-521-19797-7 .
2. ↑ Special function . In: Guido Walz (Ed.): Lexicon of Mathematics . 1st edition. Spectrum Academic Publishing House, Mannheim / Heidelberg 2000, ISBN 3-8274-0439-8 .