Asymptotics of the Mittag-Leffler function $E_a(z)$ on the negative real axis when $a\to 1$

Abstract

We consider the asymptotic expansion of the single-parameter Mittag-Leffler function $E_a(-x)$ for $x\to+\infty$ as the parameter $a\to1$. The dominant expansion when $0<a<1$ consists of an algebraic expansion of $O(x^{-1})$ (which vanishes when $a=1$), together with an exponentially small contribution that approaches $e^{-x}$ as $a\to 1$. Here we concentrate on the form of this exponentially small expansion when $a$ approaches the value 1. Numerical examples are presented to illustrate the accuracy of the expansion so obtained.