By John Heading
Since the variety of functions is gigantic, the textual content considers just a short number of themes and emphasizes the tactic itself instead of special functions. the method, as soon as derived, is proven to be one in all crucial simplicity that includes in simple terms the appliance of convinced well-defined principles. beginning with a old survey of the matter and its recommendations, matters contain the Stokes phenomenon, one and transition issues, and functions to actual difficulties. An appendix and bibliography finish the text.
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Extra resources for An Introduction to Phase-Integral Methods
This approximation is uniformly asymptotic, in the sense that the error is O(1/ h) times the main approximation, and that this is maintained throughout the domain. It should be stressed that the approximation is not valid on the boundary of the domain, and that the above discussion has not dealt with the situation further away from the transition point when other zeros of q must be taken into account. 13), consider a solution X(ξ) = e− ξ subdominant for Rl ξ > 0 in domain 1. The first correction term yields provided this correction term is much smaller in magnitude than the first approximation |e− ξ|.
It concerns problems in which q(x) is real for real x, and for which, physically speaking, energy flow is conserved along the x-axis. After Jeffreys' original paper, the need for solutions of Schrödinger's equation in wave mechanics had priority. Wentzel's  (1926) contribution should be mentioned, while it is due to Kramers  (1926) that the ‘so-called Stokes phenomenon’ came to be associated with the subject, though by name alone. Both investigators were really concerned with eigenvalue problems and potential wells.
If q = a2 − x2, the equation may be solved exactly in terms of parabolic cylinder functions. Boundary conditions are assumed such that w → 0 as x → ± ∞ ; this leads to an eigenvalue problem, the discrete values of the total energy being required in problems using Schrödinger's equation. 3) is used for both transition points x1, x2, and this leads to the formula n being an integer. Dunham  (1932) used Zwaan's method for deriving this result. J. solutions are valid between the zeros. If this is not so, this equation still gives the correct answer for the true harmonic oscillator, since exact solutions may be obtained in this particular case.