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 . For we find: (Section 2 of ) Error function: The sum of a geometric progression: Exponential function: Hyperbolic cosine: For , we have For , the integral gives, respectively: , , . The integral representation of the Mittag-Leffler function is (Section 6 of ) where the contour starts and ends at and circles around the singularities and branch points of the integrand. Related to the Laplace transform and Mittag-Leffler summation is the expression (Eq (7.5) of with ) One of the applications of the Mittag-Leffler function is in modeling fractional order viscoelastic materials. Experimental investigations into the time-dependent relaxation behavior of viscoelastic materials are characterized by a very fast decrease of the stress at the beginning of the relaxation process and an extremely slow decay for large times. It can even take a long time before a constant asymptotic value is reached. Therefore, a lot of Maxwell elements are required to describe relaxation behavior with sufficient accuracy. This ends in a difficult optimization problem in order to identify a large number of material parameters.