Second-order differential equations with fractional transition points

Olver, F. W. J.

doi:10.1090/S0002-9947-1977-0430445-7

Second-order differential equations with fractional transition points
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by F. W. J. Olver PDF

Trans. Amer. Math. Soc. 226 (1977), 227-241 Request permission

Abstract:

An investigation is made of the differential equation \[ {d^2}w/d{x^2} = \{ {u^2}{(x - {x_0})^\lambda }f(u,x) + g(u,x)/{(x - {x_0})^2}\} w,\] 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 Riekstins giving approximate solutions in terms of Bessel functions of order $1/(\lambda + 2)$ are extended and error bounds supplied.

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