The asymptotic solution of linear differential equations of the second order for large values of a parameter

Abstract

Asymptotic solutions of the differential equations d2w/dz2 = {uzn +f{z)} w 0, 1) for large positive values of u, have the formal expansions P(z) 1+ Z5=1 (f> l+ ^ M y us s 5=0 IIs where P is an exponential or Airy function for n 0 or 1 respectively. The coefficients A _s (z) and B _s (z) are given by recurrence relations. This paper proves that solutions of the differential equations exist whose asymptotic expansions in Poincaré’s sense are given by these series, and that the expansions are uniformly valid with respect to the complex variable z. The method of proof differs from those of earlier writers and fewer restrictions are made.