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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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