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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


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.