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Asymptotic expansions for the coefficient functions associated with linear second-order differential equations: The simple pole case. (English) Zbl 0709.34049
Asymptotic and computational analysis. Conference in honor of Frank W.J. Olver’s 65th birthday, Proc. Int. Symp., Winnipeg/Can. 1989, Lect. Notes Pure Appl. Math. 124, 53-73 (1990).

[For the entire collection see Zbl 0689.00009.]

The author considers uniform asymptotic expansions as n of solutions to the differential equation

d 2 w dξ 2 =(-u 2 /4ξ+(ν 2 -1)/4ξ 2 +ψ(ξ)/ξ)w;

where ν0, ψ (ξ) is holomorphic in a certain simply-connected domain Δ and ξ=0Δ. The method is described in F. W. J. Olver’s book [Asymptotics and special functions (1974; Zbl 0303.41035) ch. 12]. For the coefficients of these expansions recurrence formulas are given. Error bounds for the remainders are constructed by means of a Volterra integral equation. As an application expansions for Legendre functions P u-1/2 ν (z) and Q u-1/2 ν (z), u, ν0, Rez>0, are given.

Reviewer: E.Riekstiņs̆
MSC:
34E05Asymptotic expansions (ODE)
34A30Linear ODE and systems, general
41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
45D05Volterra integral equations