Uniform asymptotic smoothing of Stokes’s discontinuities
Uniform asymptotic smoothing of Stokes’s discontinuities
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斯托克斯不连续点的均匀渐近平滑
DOI：
10.1098/rspa.1989.0018
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发表时间：
1989
期刊：
影响因子：
--
通讯作者：
M. Berry
中科院分区：
文献类型：
--
作者：
M. Berry
在斯托克斯线上，渐近展开式中的一个指数最大地支配另一个指数，小指数的乘数迅速变化。如果展开式在其最小项附近被截断，则变化不是不连续的，而是平滑的，而且在形式上是普适的。根据奇异量F -较大和较小指数之间的差，以及斯托克斯线上的实数-乘数的变化是误差函数π-1/2 <$σ -∞ dt exp（-t2）其中σ = ImF /（2 Re F）1/2。推导过程需要控制主导级数中的指数小项;这是用丁格尔的后期项的波莱尔求和方法实现的，从最小项开始。在数值示例中，乘数是从道森积分（erfi）和第二类艾里函数（Bi）中提取的：小指数以预测的普遍方式从主导指数中出现，可以大1010倍。
Across a Stokes line, where one exponential in an asymptotic expansion maximally dominates another, the multiplier of the small exponential changes rapidly. If the expansion is truncated near its least term the change is not discontinuous but smooth and moreover universal in form. In terms of the singulant F – the difference between the larger and smaller exponents, and real on the Stokes line - the change in the multiplier is the error function π-½ ∫σ -∞ dt exp (-t2) Where σ = ImF / (2 Re F)½. The derivation requires control of exponentially small terms in the dominant series; this is achieved with Dingle’s method of Borel summation of late terms, starting with the least term. In numerical illustrations the multiplier is extracted from Dawson’s integral (erfi) and the Airy function of the second kind (Bi): the small exponential emerges in the predicted universal manner from the dominant one, which can be 1010 times larger.