Mittag-Leffler function
The Mittag-Leffler function is a mathematical function named after the mathematician Magnus Gösta Mittag-Leffler , which appears in the solutions of certain fractional integral equations (e.g. when investigating random movements or Lévy flights ). It is given by
- ,
where is the gamma function . The series converges for all with a positive real part . In the special case , the exponential function results .
The generalized Mittag-Leffler function describes an interpolation between exponential and polynomial behavior and is given by
- .
Special cases of this function are
- Gaussian error function:
- Hyperbolic Sinus:
literature
- MG Mittag-Leffler: Sur la nouvelle fonction E_alpha (x). In: Comptes Rendus de l'Académie des sciences 137/1903, pp. 554-558
- RK Saxena, AM Mathai HJ Haubold: On Fractional Kinetic Equations. In: Astrophysics & Space Science 282/2002, pp. 281–287 ( ISSN 0004-640X ), ( pdf version )
Web links
- Eric W. Weisstein : Mittag-Leffler Function . In: MathWorld (English).