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

${\displaystyle \mathrm {E} _{\alpha }(x)=\sum _{k=0}^{\infty }{\frac {x^{k}}{\Gamma (\alpha k+1)}}}$,

where is the gamma function . The series converges for all with a positive real part . In the special case , the exponential function results . ${\displaystyle \Gamma (x)}$ ${\displaystyle \alpha }$ ${\displaystyle \alpha =1}$

The generalized Mittag-Leffler function describes an interpolation between exponential and polynomial behavior and is given by

${\displaystyle \mathrm {E} _{\alpha ,\beta }(x)=\sum _{k=0}^{\infty }{\frac {x^{k}}{\Gamma (\alpha k+\beta )}}}$.

Special cases of this function are

• Gaussian error function:

${\displaystyle E_{1/2,1}(z)=\exp(z^{2})\operatorname {erfc} (-z)}$

• Hyperbolic Sinus:

${\displaystyle E_{2,1}(z)=\cosh({\sqrt {z}})}$

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).