Special functions
In calculus, a branch of mathematics is called certain functions as special functions, because ( for example, in mathematical physics ) play a major role in both mathematics itself and in its applications. Most special functions are holomorphic or meromorphic and can be developed in series. They are called, among other reasons as special because they are in many ways interrelated. In its investigation, such relationships are described and new found. Some special features include the transcendental functions and are also called higher transcendental functions because of their special position.
Since the 19th century, various approaches have been developed, important special functions can be treated as special cases of closed representable function droves with them. These include, inter alia, the Meijersche G function, Foxsche H function and the hypergeometric function.
List of some special functions
- The following special functions are generalizations of the faculty or of the gamma function gamma function
- Beta function
- Pochhammer symbol
- Polygamma functions ( special cases: Digammafunktion, Trigammafunktion )
- Barnes'sche G function
- Special functions arising from the hypergeometric function for specific parameter Legendre polynomials
- Hermitian polynomials
- Laguerre polynomials
- Bessel functions
- Polylogarithms
- Nielsen function
- Clausen functions
- Riemann zeta function and Riemann Xi - function
- Elliptic integral functions
- Dirichlet eta function
- Dirichlet lambda function
- Dirichlet beta function
- Airy function
In the multivariate analysis, special features in several (usually complex ) variables are used.
- Special functions in several parameters Generalized hypergeometric functions Appellsche functions
More Special functions of theoretical physics:
- Clebsch -Gordan symbols
- Wigner nj symbols
Areas of application
Many of these functions are solutions of differential equations that occur in important application scenarios. Special functions are also the backbone of many calculations with computer algebra systems (Mathematica, Maple, ...).
More recently, the properties of special functions with the help of computer algebra and " symbolic computation " are investigated. In analytical number theory are of particular importance.