The asymptotic solution of linear differential equations of the second order in a domain containing one transition point
Frank W. J. Olver · Philosophical Transactions of the Royal Society of London Series A Mathematical and Physical Sciences · 1956
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.