Numerical Algorithm for Calculating the Generalized Mittag-Leffler Function

Abstract

A numerical algorithm for calculating the generalized Mittag-Leffler $\mathrm{E}_{\alpha,\beta}(z)$ function for arbitrary complex argument $z$ and real parameters $\alpha>0$ and $\beta\in\mathbb{R}$ is presented. The algorithm uses the Taylor series, the exponentially improved asymptotic series, and integral representations to obtain optimal stability and accuracy of the algorithm. Special care is applied to the limits of validity of the different schemes to avoid instabilities in the algorithm.