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Locating the Zeros of Partial Sums of ez with Riemann-Hilbert Methods

T. Kriecherbauer
Fakultät für Mathematik, Universität Bochum, Germany

A. B. J. Kuijlaars
Department of Mathematics, Katholieke Universiteit Leuven, Belgium

K. T.-R. McLaughlin
Department of Mathematics, University of Arizona, Tucson

P. D. Miller
Department of Mathematics, University of Michigan, Ann Arbor

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 pn-1(z)=1+z+z2/2!+…+zn-1/(n-1)!. Our proof is based on a representation of pn-1(nz) in terms of an integral of the form ∫enφ(s)(s-z)-1ds. 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 φ.