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Historical Survey of the Problem
The W K B J Solutions
The Stokes Phenomenon
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Airy equation Airy function Airy integral anti-Stokes line approximate solutions asymptotic expansions asymptotic expressions asymptotic series asymptotic solutions Bessel's equation boundary branch cut comparison equation complex z-plane connection formulae considered correction term denote derived differential equation domain of validity dominant coefficient dominant solution dominant term eigenvalues energy flow error term exact solution exponential factor further zeros given Hankel functions harmonic oscillator Hence incident wave ionosphere Jeffreys 66 Langer line argz linear magnitude mathematical matrix medium namely negative real axis obtain occurs parabolic cylinder functions phase phase-integral methods point of order positive real axis potential barrier problem real axis reflection coefficient replaced represents a wave section 2.4 solution round Stokes constants Stokes line Stokes phenomenon subdominant solution subdominant term theory Tracing the solution transition point uniformly asymptotic vanishes variable W.K.B.J. approximations W.K.B.J. solutions x-axis yielding zeros of q