Computation of the generalized Mittag-Leffler function and its inverse in the complex plane

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 −∞+ (2k − 1)π i for α → 1− and towards −∞+ 2kπ i for α → 1+ (k ∈ Z). All the 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.