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Zbl 0591.34048
Boyd, W.G.C.; Dunster, T.M.
Uniform asymptotic solutions of a class of second-order linear differential equations having a turning point and a regular singularity, with an application to Legendre functions. (English)
[J] SIAM J. Math. Anal. 17, 422-450 (1986). ISSN 0036-1410; ISSN 1095-7154/e

Summary: The asymptotic behaviour, as a parameter $u\to \infty$, of solutions of second-order linear differential equations with a turning point and a regular (double pole) singularity is considered. It is shown that the solutions can be approximated by expressions involving Bessel functions in a region which includes both the turning point and the singularity. Explicit error bounds for the difference between the approximations and the exact solutions are established. The theory is applied to find uniform asymptotic expansions for Legendre functions.

MSC 2000:: *34E05 Asymptotic expansions (ODE)
34A30 Linear ODE and systems
34C05 Qualitative theory of some special solutions of ODE

Keywords: regular singularity; second-order linear differential equations; turning point

Cited in: Zbl 1055.33008 Zbl 0982.33003 Zbl 0606.33011

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