Computation of the Generalized Mittag-Leffler Function and its Inverse in the Complex Plane

Abstract.

The generalized Mittag-Leffler function Eα,β⁢z has been studied for arbitrary complex argument z∈C and parameters α∈R+ and β∈R. This function plays a fundamental role in the theory of fractional differential equations and numerous applications in physics. The Mittag-Leffler function interpolates smoothly between exponential and algebraic functional behaviour. A numerical algorithm for its evaluation has been developed. The algorithm is based on integral representations and exponential asymptotics. Results of extensive numerical calculations for Eα,β⁢z in the complex z-plane are reported here. We find that all complex zeros emerge from the point z=1 for small α. They diverge towards -∞+2⁢k-1⁢π⁢i for α→1- and towards -∞+2⁢k⁢π⁢i for α→1+ (k∈Z). All complex zeros collapse pairwise onto the negative real axis for α→2. We introduce and study also the inverse generalized Mittag-Leffler function Lα,β⁢z defined as the solution of the equation Lα,β⁢Eα,β⁢z=z. We determine its principal branch numerically.