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 ${\int }_{\gamma }\frac{e^{n\varphi (s)}}{s-z}ds$. 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 $\varphi$.

arXiv comment13 pages, 3 figures; to appear in proceedings of "Integrable Systems, Random Matrices, and Applications, a conference in honor of Percy Deift's 60th birthday"

Cited by in corpus (1)

• Dynamic behavior of the roots of the Taylor polynomials of the Riemann xi function with growing degree