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

The generalized Mittag-Leffler function E-alpha,E-beta (z) has been studied for arbitrary complex argument z is an element of C and parameters alpha is an element of R+ and beta is an element of 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 E-alpha,E-beta (z) in the complex z -plane are reported here. We find that all complex zeros emerge from the point z =1 for small alpha. They diverge towards -infinity+(2 k -1)pi i for alpha -> 1(-) and towards -infinity+2 k pi i for alpha -> 1(+) ( k is an element of Z). All the complex zeros collapse pairwise onto the negative real axis for alpha -> 2. We introduce and study also the inverse generalized Mittag-Leffler function L-alpha,L-beta (z) defined as the solution of the equation L-alpha,L-beta (E-alpha,E-beta (z )) = z . We determine its principal branch numerically.