Numerische Mathematik

, Volume 91, Issue 1, pp 147–193 | Cite as

Uniform asymptotic expansion of $J_\nu(\nu a)$ via a difference equation

  • Z. Wang
  • R. Wong
Original article

Summary.

There are two ways of deriving the asymptotic expansion of \(J_\nu(\nu a)\), as \(\nu \to \infty\), which holds uniformly for \(a\geq 0\). 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 \(J_{\nu+1}(x)+J_{\nu-1}(x)=(2\nu /x)J_\nu(x)\). Our approach will lead to a satisfactory development of a turning point theory for second-order linear difference equations.

Mathematics Subject Classification (1991): 65D20 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Z. Wang
    • 1
  • R. Wong
    • 1
  1. 1.Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong, China; e-mail: mawong@cityu.edu.hk HK

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