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Zbl 0798.34083
Dunster, T.M.; Lutz, D.A.; Schäfke, R.
Convergent Liouville-Green expansions for second-order linear differential equations, with an application to Bessel functions. (English)
[J] Proc. R. Soc. Lond., Ser. A 440, No.1908, 37-54 (1993). ISSN 0080-4630

The paper deals with second-order linear differential equations in some complex domain of the form $d\sp 2W/d \xi\sp 2 = (u\sp 2 + \psi (\xi))W$ where $u$ is a large complex parameter and $\psi$ is an analytic function. Under certain mild assumption on $\psi$, the authors are able to obtain convergent Liouville-Green expansions for the solutions in terms of inverse factorials using Laplace transforms and the integral equation method or, alternatively, the inverse Laplace transform. These expansions converge for ${\germ R} (u)>0$, uniformly for $\xi$ in a certain subdomain of the domain of asymptotic validity. Finally, the general result is used to derive convergent Liouville-Green expansions for the modified Bessel functions $I\sb \nu (\nu z)$ and $K\sb \nu (\nu z)$ of large order $\nu$.
[Ch.Tretter (Regensburg)]

MSC 2000:: *34L10 Eigenfunction expansions, etc. (ODE)
33C10 Cylinder functions, etc.

Keywords: second-order linear differential equations; complex domain; Liouville- Green expansions; inverse factorials; Laplace transforms; integral equation method; inverse Laplace transform; modified Bessel functions

Cited in: Zbl 1017.34089

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