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

R. Hilfer{}^{{1,2}} and H.J. Seybold{}^{1}
{}^{1}Institut für Computerphysik, Universität Stuttgart, 70569 Stuttgart
{}^{2}Institut für Physik, Universität Mainz, 55099 Mainz, Germany

The generalized Mittag-Leffler function \mbox{\rm E}_{{\alpha,\beta}}(z) has been studied for arbitrary complex argument z\in\mathbb{C} and parameters \alpha\in\mathbb{R}^{+} and \beta\in\mathbb{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 \mbox{\rm E}_{{\alpha,\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 -\infty+(2k-1)\pi{\rm i} for \alpha\to 1^{-} and towards -\infty+2k\pi{\rm i} for \alpha\to 1^{+} (k\in\mathbb{Z}). All complex zeros collapse pairwise onto the negative real axis for \alpha\to 2. We introduce and study also the inverse generalized Mittag-Leffler function \mbox{\rm L}_{{\alpha,\beta}}(z) defined as the solution of the equation \mbox{\rm L}_{{\alpha,\beta}}(\mbox{\rm E}_{{\alpha,\beta}}(z))=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