Second-order linear differential equations with two turning points

Abstract
Differential equations of the form d 2 w / d x 2 = { u 2 f ( u , a , x ) + g ( u , a , x ) } w are considered for large values of the real parameter u . Here x is a real variable ranging over an open, possibly infinite, interval ( x 1 , x 2 ), and a is a bounded real parameter. It is assumed that f {u, a, x) and g{u,a, x) are free from singularity within ( x 1 , x 2 ), and f (u, a, x) has exactly two zeros, which depend continuously on a and coincide for a certain value of a . Except in the neighbourhoods of the zeros, g(u,a,x) is small in absolute value compared with u 2 f(u, a, x ). By application of the Liouville transformation, the differential equation is converted into one of four standard forms, with continuous coefficients. Asymptotic approximations for the solutions are then constructed in terms of parabolic cylinder functions. These approximations are valid for large u , uniformly with respect to x ε ( x 1 , x 2 ) and also uniformly with respect to a . Each approximation is accompanied by a strict and realistic error bound. The paper also includes some new properties of parabolic cylinder functions.

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