The asymptotic solution of linear differential equations of the second order in a domain containing one transition point

Abstract

In a recent paper (Olver 1954) I considered asymptotic expansions of solutions of the differential equation dz2 {up(z)+q(z)}w, ( M ) for large positive values of the parameter u , which are uniform with respect to z when z lies in a simply-connected domain D in which p(z) , q(z) are analytic functions of z independent of u .Three cases A, B and C were considered. In case A the equation had no transition points in D. In case B there was one transition point in D, a simple zero of p(z) . In case C there was again one transition point, this time a double pole of p(z) . It was shown that case C can be transformed into case A.