Locating the zeros of partial sums of exp(z) with Riemann-Hilbert methods

Details Locating the zeros of partial sums of exp(z) with Riemann-Hilbert methods Locating the zeros of partial sums of exp(z) with Riemann-Hilbert methods

2007-09-08

arxiv.org/abs/0709.1213v1

Contemporary Mathematics 458 (2008), 183--196

Mathematics

Classical Analysis and ODEs

13 pages, 3 figures; to appear in proceedings of "Integrable Systems, Random Matrices, and Applications, a conference in honor

Scientific paper

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 $\int_{\gamma} \frac{e^{n\phi(s)}}{s-z}ds$. 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 $\phi$.

Affiliated with

Also associated with

No associations

Rating

Locating the zeros of partial sums of exp(z) with Riemann-Hilbert methods does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Locating the zeros of partial sums of exp(z) with Riemann-Hilbert methods, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Locating the zeros of partial sums of exp(z) with Riemann-Hilbert methods will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-44456