Exponential Asymptotics of the Mittag—Leffler Function

Abstract. The Stokes lines/curves are identified for the Mittag—Leffler function $$ E_{\alpha, \beta}(z)=\sum^{\infty}_{n=0}\frac{z^n}{\Gamma(\alpha n+\beta)},\qquad\mathop{\rm Re}\nolimits \:\alpha > 0. $$ 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.