Sie sind hier: ICP » R. Hilfer » Publikationen

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

R. Hilfer1,2 and H.J. Seybold1
1Institut für Computerphysik, Universität Stuttgart, 70569 Stuttgart
2Institut für Physik, Universität Mainz, 55099 Mainz, Germany

The generalized Mittag-Leffler function Eα,βz has been studied for arbitrary complex argument zC and parameters αR+ and βR. 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,βz 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-1πi for α1- and towards -+2kπi for α1+ (kZ). All complex zeros collapse pairwise onto the negative real axis for α2. We introduce and study also the inverse generalized Mittag-Leffler function Lα,βz defined as the solution of the equation Lα,βEα,βz=z. We determine its principal branch numerically.

Key words and phrases:
special functions of mathematical physics, fractional calculus, generalized Mittag-Leffler functions, numerical algorithms
PACS: 02.30.Gp, 02.60.Gf