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# Second-Order Differential Equations with Fractional Transition Points

F. W. J. Olver
Transactions of the American Mathematical Society
Vol. 226 (Feb., 1977), pp. 227-241
DOI: 10.2307/1997952
Stable URL: http://www.jstor.org/stable/1997952
Page Count: 15
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## Abstract

An investigation is made of the differential equation $$d^2w/dx^2 = \{u^2(x - x_0)^\lambda f(u, x) + g(u, x)/(x - x_0)^2\}w$$, in which $u$ is a large real (or complex) parameter, $\lambda$ is a real constant such that $\lambda > -2, x$ is a real (or complex) variable, and $f(u, x)$ and $g(u, x)$ are continuous (or analytic) functions of $x$ in a real interval (or complex domain) containing $x_0$. The interval (or domain) need not be bounded. Previous results of Langer and Riekstiņš giving approximate solutions in terms of Bessel functions of order $1/(\lambda + 2)$ are extended and error bounds supplied.

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