Uniform asymptotic expansion of Jν (νa) via a difference equation

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Original languageEnglish
Pages (from-to)147-193
Journal / PublicationNumerische Mathematik
Volume91
Issue number1
Publication statusPublished - Mar 2002

Abstract

There are two ways of deriving the asymptotic expansion of Jν(νa), as ν → ∞, which holds uniformly for a ≥ 0. One way starts with the Bessel equation and makes use of the turning point theory for second-order differential equations, and the other is based on a contour integral representation and applies the theory of two coalescing saddle points. In this paper, we shall derive the same result by using the three term recurrence relation Jν+1 (x) + Jν-1(x) = (2ν/x) Jν(x). Our approach will lead to a satisfactory development of a turning point theory for second-order linear difference equations.

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