On a uniform treatment of Darboux's method
type: Item
instance of: scholarly article
title: On a uniform treatment of Darboux's method
zbMATH Open document ID: 1074.41505
author: Yuqiu Zhao
author: Roderick Wong
published in: Constructive Approximation
publication date: 2005-04-15T00:00:00Z
review text: Given an analytic function \(F(z)\) inside the unit circle \(| z|<1\), the authors propose a new method to approximate the late coefficients \(a_n\) of the Maclaurin expansion \(F(z)=\sum_{n=0}^\infty a_nz^n\) when \(F(z)\) has only a finite number of algebraic singularities on the unit circle \(| z|=1\). It is well known that the Darboux method gives an asymptotic expansion of \(a_n\) when \(n\to\infty\). But this method does not work when the singularities of \(F(z)\) on the unit circle \(| z|=1\) approach each other, that is, the method is not uniform in the parameter which controls the confluence of singularities. A uniform method is known since 1968 for a special case of confluence, although it is too complicated. The authors propose in this paper a more simple uniform method somehow inspired in the previous work of Chester, Friedman, Ursell, Blestein, Olde Daalhuis and Temme. In the case of two coalescing singularities, the authors obtain a uniform expansion of \(a_n\) for large \(n\) in terms of two sequences of inverse powers of \(n\), each sequence multiplied by a Bessel functions of the first kind. In the general case of \(p\) coalescing singularities, the uniform expansion of \(a_n\) is given in terms of \(p\) sequences of inverse powers of \(n\), each sequence multiplied by more complicated functions than the Bessel functions. The method is applied to the ultraspherical polynomials \(P_n^{(\lambda)}(x)\). Updated 2.2.2006
Mathematics Subject Classification ID: 41A60
Mathematics Subject Classification ID: 33C10
Mathematics Subject Classification ID: 33E20
zbMATH DE Number: 2157507
zbMATH Keywords: coalescing algebraic singularities
zbMATH Keywords: uniform asymptotic expansion
zbMATH Keywords: Darboux's method
reviewed by: José L. López
MaRDI profile type: Publication
full work available at URL: s00365-004-0562-9
OpenAlex ID: W1988862537
DOI: 10.1007/S00365-004-0562-9
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description: scientific article; zbMATH DE number 2157507