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)πi for α→1− and towards −∞+2kπi 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.