LOCATING THE ZEROS OF PARTIAL SUMS OF e WITH RIEMANN-HILBERT METHODS
نویسنده
چکیده
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) = Pn−1 k=0 z/ k! . Our proof is based on a representation of pn−1(nz) in terms of an integral of the form R γ e 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 RiemannHilbert analysis in particular for points z that are close to the critical points of φ.
منابع مشابه
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