用Riemann-Hilbert方法求exp（z）部分和的零点
Locating the zeros of partial sums of exp(z) with Riemann-Hilbert methods
摘要：
本文给出了指数级数部分和的一致渐近展开。我们表明如何将这些信息用于在以后的出版物中获得完整的和明确的可计算的渐近展开与误差界的泰勒多项式$p_{n-1}(z) = \sum_{k=0}^{n-1} z^k/ k!$的所有零。我们的证明是基于一个表示$p_{n-1}(nz)$的积分形式$\int_γ \frac{e^{nϕ(s)}}{s-z}ds$。我们演示了如何使用黎曼-希尔伯特方法得到这样的积分的一致展开。与经典的最速下降分析的比较显示了Riemann-Hilbert分析的优点，特别是对于接近$ϕ$的临界点的点$z$。
英文摘要：
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 $\int_γ \frac{e^{nϕ(s)}}{s-z}ds$. 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 $ϕ$.
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