*(English)*Zbl 1096.65024

Summary: The generalized Mittag-Leffler function ${E}_{\alpha ,\beta}\left(z\right)$ has been studied for arbitrary complex argument $z\in \u2102$ 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 ${E}_{\alpha ,\beta}\left(z\right)$ 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 ${L}_{\alpha ,\beta}\left(z\right)$ defined as the solution of the equation ${L}_{\alpha ,\beta}\left({E}_{\alpha ,\beta}\left(z\right)\right)=z$. We determine its principal branch numerically.