Numerical Evaluation of Mittag-Leffler Functions

TL;DR

This paper presents a numerical method for evaluating Mittag-Leffler functions using contour integrals, comparing different contours and addressing pole handling, with additional discussion on rational approximation on the negative real axis.

Contribution

It introduces a contour integral approach with a novel pole handling technique and compares parabolic and hyperbolic contours for efficient computation.

Findings

Contour integral methods effectively evaluate Mittag-Leffler functions.

Handling poles in the integrand improves numerical accuracy.

Rational approximation is viable on the negative real axis.

Abstract

The Mittag-Leffler function is computed via a quadrature approximation of a contour integral representation. We compare results for parabolic and hyperbolic contours, and give special attention to evaluation on the real line. The main point of difference with respect to similar approaches from the literature is the way that poles in the integrand are handled. Rational approximation of the Mittag-Leffler function on the negative real axis is also discussed.