Special functions | Analytic functions
Mittag-Leffler function
In mathematics, the Mittag-Leffler function is a special function, a complex function which depends on two complex parameters and . It may be defined by the following series when the real part of is strictly positive: where is the gamma function. When , it is abbreviated as .For , the series above equals the Taylor expansion of the geometric series and consequently . In the case and are real and positive, the series converges for all values of the argument , so the Mittag-Leffler function is an entire function. This function is named after Gösta Mittag-Leffler. This class of functions are important in the theory of the fractional calculus. For , the Mittag-Leffler function is an entire function of order , and is in some sense the simplest entire function of its order. The Mittag-Leffler function satisfies the recurrence property (Theorem 5.1 of ) from which the Poincaré asymptotic expansion follows, which is true for . (Wikipedia).
