Locating the zeros of partial sums of exp(z) with Riemann-Hilbert methods

Abstract

In this paper we derive uniform asymptotic expansions for the partial sums of the exponential series. We indicate how this information will be used in a later publication to obtain full and explicitly computable asymptotic expansions with error bounds for all zeros of the Taylor polynomials $p_{n-1}(z) = \sum_{k=0}^{n-1} z^k/ k!$. Our proof is based on a representation of $p_{n-1}(nz)$ in terms of an integral of the form $\int_{\gamma} \frac{e^{n\phi(s)}}{s-z}ds$. We demonstrate how to derive uniform expansions for such integrals using a Riemann-Hilbert approach. A comparison with classical steepest descent analysis shows the advantages of the Riemann-Hilbert analysis in particular for points $z$ that are close to the critical points of $\phi$.