Computing the Hilbert transform of the generalized Laguerre and Hermite weight functions
Computing the Hilbert transform of the generalized Laguerre and Hermite weight functions
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DOI：
10.1023/a:1021915128433
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发表时间：
2001-09-01
期刊：
影响因子：
1.5
通讯作者：
Waldvogel, J
中科院分区：
文献类型：
--
作者：
Gautschi, W;Waldvogel, J
给出了Hilbert变换积分w（t）dt/（t-x）的显式表达式，其中w是广义Laguerre权函数w（t）= 0（当t小于或等于0时），w（t）= t（o）e（-t）（当0 < t < -无穷大，α>-1，x > 0时），或Hermite权函数w（t）= e（-t2），-无穷大< t <无穷大，并且-infinity < x < infinity。此外，数值方法的评估讨论的基础上递归，轮廓积分和鞍点，渐近，级数展开。本文还研究了积分pi（n）（t;w）w（t）dt/（t-x），n= 0，1，2，…，其中pi（n）（t;w）是广义Laguerre，分别为Hermite，n次多项式。
Explicit formulae are given for the Hilbert transform integral w(t)dt/(t-x), Where w is either the generalized Laguerre weight function w(t) = 0 if t less than or equal to 0, w(t) = t(o)e(-t) if 0 < t < -infinity, and alpha> -1, x > 0, or the Hermite weight function w(t) = e(-t 2), -infinity < t < infinity, and -infinity < x < infinity. Furthermore, numerical methods of evaluation are discussed based on recursion, contour integration and saddle-point, asymptotics, and series expansions. We also study the numerical stability of the three-term recurrence relation satisfied by the integrals integral pi (n)(t;w)w(t)dt/(t-x), n=0,1,2,...,where pi (n)(t;w) is the generalized Laguerre, resp. the Hermite, polynomial of degree n.