Phase-Integral Methods I

An arbitrary but finite number of transition points of varying multiplicities is postulated for a second order linear differential equation. The approximate phase-integral solutions (‘W.K.B.J.’ solutions) are not valid in the neighbourhood of any transition point; the domain of uniform validity of any particular approximate solution in the complex plane of the independent variable is examined, and rules delineating such a domain are derived in terms of anti-Stokes lines radiating out from all the transition points. More general solutions of the equation are found, uniformly valid in a domain containing a chosen number of transition points; the extent of this domain is examined when traced up to the neighbourhood of the remaining transition points.