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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 $z\in\bbfC$ and parameters $\alpha\in\bbfR^+$ and $\beta\in\bbfR$. 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 $\alpha \to 1^+$ $(k\in\bbfZ)$. 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.

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