Uniform Asymptotic Solution of Second Order Linear Differential Equations without Turning Varieties

The purpose of this paper is to initiate the study of a new kind of asymptotic series expansion for solutions of differential equations containing a parameter. We obtain uniform asymptotic solutions for certain equations of the form ê»y" = ait, e)y , ( )' = d/dt, where n is a positive integer, t and e are real variables ranging over \t\ S U, 0 < e ^ €o, and a is a function infinitely differentiable on the closure of this domain. We require that a(i, e) satisfy conditions which can be regarded as generalized nonturning-point conditions. These conditions imply the absence of secondary turning points, and reduce in the simplest case to the condition a(¿, 0) 5¿ 0, but also include cases (the interesting ones) in which a(0, 0) = 0. |