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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 $n\to \infty$ of solutions to the differential equation

$\frac{{d}^{2}w}{d{\xi }^{2}}=\left(-{u}^{2}/4\xi +\left({\nu }^{2}-1\right)/4{\xi }^{2}+\psi \left(\xi \right)/\xi \right)w;$

where $\nu \ge 0$, $\psi$ ($\xi$) is holomorphic in a certain simply-connected domain ${\Delta }$ and $\xi =0\in {\Delta }$. 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 ${P}_{u-1/2}^{\nu }\left(z\right)$ and ${Q}_{u-1/2}^{\nu }\left(z\right)$, $u\to \infty$, $\nu \ge 0$, $Rez>0$, are given.

Reviewer: E.Riekstiņs̆
##### MSC:
 34E05 Asymptotic expansions (ODE) 34A30 Linear ODE and systems, general 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.) 45D05 Volterra integral equations