Exponential Asymptotics of the Mittag—Leffler Function

The Stokes lines/curves are identified for the Mittag—Leffler function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ E_{\alpha, \beta}(z)=\sum^{\infty}_{n=0}\frac{z^n}{\Gamma(\alpha n+\beta)},\qquad\mathop{\rm Re}\nolimits \:\alpha > 0. $$ \end{document} When α is not real, it is found that the Stokes curves are spirals. Away from the Stokes lines/curves, exponentially improved uniform asymptotic expansions are obtained. Near the Stokes lines/curves, Berry-type smooth transitions are achieved via the use of the complementary error function.