### R. Hilfer, H.J. Seybold

Integral Transforms and Special Functions **17**, 637 (2006)

https://doi.org/10.1080/10652460600725341

submitted on

Monday, March 21, 2005

The generalized Mittag-Leffler function Eα,β (z) has been studied for arbitrary complex argument and real parameters. 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 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 negative infinity for alpha approaching unity. All the complex zeros collapse pairwise onto the negative real axis for α approaching 2. We introduce and study also the inverse generalized Mittag-Leffler function. We determine its principal branch numerically.

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