The calculation 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.

Keywords:

• Mittag-Leffler function
• calculation Mittag-Leffler function
• Integral representation

AMS Classification:

• 33E12

Acknowledgments

The author thanks M. Yu. Dudikov for the translation of the paper into English.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the Russian Foundation for Basic Research (RFBR) (projects No 19-44-730005, 20-07-00655).