##
Locating the Zeros of Partial Sums of *e*^{z} with Riemann-Hilbert Methods

^{z}

**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 *p _{n-1}(z)=1+z+z^{2}/2!+…+z^{n-1}/(n-1)!*. Our proof is based on a representation of

*p*in terms of an integral of the form

_{n-1}(nz)*∫e*. 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

^{nφ(s)}(s-z)^{-1}ds*z*that are close to the critical points of

*φ*.