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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 $$ \frac{d\sp 2w}{d\xi\sp 2}=(- u\sp 2/4\xi +(\nu\sp 2-1)/4\xi\sp 2+\psi (\xi)/\xi)w; $$ 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 {\it 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 $P\sp{\nu}\sb{u-1/2}(z)$ and $Q\sp{\nu}\sb{u-1/2}(z)$, $u\to \infty$, $\nu\ge 0$, $Rez>0$, are given.
Reviewer: E.Riekstiņš
34E05Asymptotic expansions (ODE)
34A30Linear ODE and systems, general
41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
45D05Volterra integral equations