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Computation of the generalized Mittag-Leffler function and its inverse in the complex plane. (English) Zbl 1096.65024

Summary: The generalized Mittag-Leffler function E α,β (z) has been studied for arbitrary complex argument z and parameters α + and β. 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 is developed. The algorithm is based on integral representations and exponential asymptotics. Results of extensive numerical calculations for E α,β (z) in the complex z-plane are reported. We find that all complex zeros emerge from the point z=1 for small α. They diverge towards -+(2k-1)πi for α1 - and towards -+2kπi for α1 + (k). All the complex zeros collapse pairwise onto the negative real axis for α2. We introduce and study also the inverse generalized Mittag-Leffler function L α,β (z) defined as the solution of the equation L α,β (E α,β (z))=z. We determine its principal branch numerically.

MSC:
65D20Computation of special functions, construction of tables
33E12Mittag-Leffler functions and generalizations
33F05Numerical approximation and evaluation of special functions