# ML

Numerical evaluation of two and three parameter Mittag-Leffler functions. The Mittag-Leffler function plays a fundamental role in fractional calculus. In the present paper, a method is introduced for the efficient computation of the Mittag-Leffler function based on the numerical inversion of its Laplace transform. The approach taken is to consider separate regions in which the Laplace transform is analytic and to look for the contour and discretization parameters allowing one to achieve a given accuracy. The optimal parabolic contour algorithm selects the region in which the numerical inversion of the Laplace transform is actually performed by choosing the one in which both the computational effort and the errors are minimized. Numerical experiments are presented to show accuracy and efficiency of the proposed approach. An application to the three parameter Mittag-Leffler function is also presented.