A complete asymptotic expansion of the Jacobi functions with error bounds
type: Item
instance of: scholarly article
title: A complete asymptotic expansion of the Jacobi functions with error bounds
zbMATH Open document ID: 0631.33006
DOI: 10.1007/BF02008376
author: Xiuchun Li
published in: Acta Mathematicae Applicatae Sinica. English Series
publication date: 1987-01-01T00:00:00Z
review text: Jacobi functions are in fact Gaussian hypergeometric functions of the form \({}_ 2F_ 1((\alpha +\beta +1-i\lambda),(\alpha +\beta +1+i\lambda);\alpha +1;-\sinh^ 2t)\). The asymptotic expansion of this function is given as \(\lambda\to \infty\), uniformly with respect to \(t\in [0,T]\), where T is a fixed positive parameter. Especially near the origin a Bessel function is needed as basic approximant. The complete expansion is given; for the remainder a strict bound is given together with an order estimate. This shows the asymptotic nature of the expansion. The error bound is obtained by using the Gronwall-Bellman inequality, which is applied to an integral equation for the remainder.
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 41A60
zbMATH DE Number: 4025843
zbMATH Keywords: error bounds
zbMATH Keywords: Jacobi functions
reviewed by: Nico M. Temme
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4025843