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

Abstract

We consider the asymptotic expansion of the one-parameter Mittag-Leffler function \(E_a(-x)\) for \(x\rightarrow +\infty \) as the parameter \(a\rightarrow 1\). 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 \(a\rightarrow 1\). 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.