The calculation of the Mittag-Leffler function

TL;DR

This paper develops new integral representations for the Mittag-Leffler function that are suitable for numerical computation, and verifies their accuracy through comparison with known formulas.

Contribution

It introduces novel integral representations of the Mittag-Leffler function that eliminate complex variables, facilitating numerical calculation and verification.

Findings

New integral representations match known functions accurately.

Numerical calculations confirm the correctness of the new formulas.

The approach simplifies the computation of the Mittag-Leffler function.

Abstract

The problem of calculating the Mittag-Leffler function $E_{\rho,\mu} (z)$ is considered in the paper. To solve this problem integral representations for the function $E_{\rho,\mu}(z)$ are transformed in such a way that they could not contain complex variables and parameters. Integral representations written in this form allow one to use standard methods of numerical integration to calculate integrals contained in them. To verify the correctness of the integral representations obtained the function $E_{\rho,\mu}(z)$ was calculated both with the use of obtained formulas and with the use of known representations of the Mittag-Leffler function. The calculation results demonstrate their exact matching. This fact is indicative of the correctness of new integral representations of the function $E_{\rho,\mu}(z)$ that were obtained.