Stieltjes变换和Stokes现象

W. Boyd
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引用次数: 42

摘要

最近,Berry, Olver和Jones发现了指数小剩余项的一致渐近展开式,当渐近展开式被最优截断时产生。这些均匀的膨胀描述了当斯托克斯线被越过时,剩余物行为的快速变化。当一个函数可以用Stieltjes变换表示时,我们将展示如何找到这样的一致展开。这种方法的优点是:直接计算均匀展开,容易找到远离Stokes线的非均匀展开,并且可以建立明确的误差界限。我们通过对修正贝塞尔函数Kv(z)的应用来说明该方法。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Stieltjes transforms and the Stokes phenomenon
Recently, Berry, Olver and Jones have found uniform asymptotic expansions for the exponentially small remainder terms that result when asymptotic expansions are optimally truncated. These uniform expansions describe the rapid change in the behaviour of the remainders as a Stokes line is crossed. We show how such uniform expansions may be found when a function can be expressed as a Stieltjes transform. Such an approach has the following advantages: the uniform expansion is calculated directly, non-uniform expansions away from the Stokes line are readily found, and explicit error bounds may be established. We illustrate the method by application to the modified Bessel function Kv(z).
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