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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 $$E_{\alpha,\beta}(z)$$ 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 is developed. The algorithm is based on integral representations and exponential asymptotics. Results of extensive numerical calculations for $$E_{\alpha,\beta} (z)$$ in the complex $$z$$-plane are reported. We find that all complex zeros emerge from the point $$z=1$$ for small $$\alpha$$. They diverge towards $$-\infty+ (2k-1)\pi i$$ for $$\alpha\to 1^-$$ and towards $$-\infty+2k\pi i$$ for $$\alpha \to 1^+$$ $$(k\in\mathbb{Z})$$. All the complex zeros collapse pairwise onto the negative real axis for $$\alpha\to 2$$. We introduce and study also the inverse generalized Mittag-Leffler function $$L_{\alpha,\beta}(z)$$ defined as the solution of the equation $$L_{\alpha,\beta}(E_{\alpha, \beta}(z))=z$$. We determine its principal branch numerically.

MSC:
 65D20 Computation of special functions and constants, construction of tables 33E12 Mittag-Leffler functions and generalizations 33F05 Numerical approximation and evaluation of special functions
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References:
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