By John Heading
Since the variety of purposes is gigantic, the textual content considers just a short number of issues and emphasizes the tactic itself instead of particular purposes. the method, as soon as derived, is proven to be one in all crucial simplicity that contains in simple terms the applying of convinced well-defined ideas. beginning with a ancient survey of the matter and its suggestions, topics contain the Stokes phenomenon, one and transition issues, and purposes to actual difficulties. An appendix and bibliography finish the text.
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Additional info for An Introduction to Phase-Integral Methods
Phase-integral methods are applied to the particular equation in which h is a large parameter and q(x, h) a variable function of x and h. The real independent variable x may often be replaced by the complex variable z. Under certain circumstances it will be shown that approximate solutions of this equation are the error term of the form shown being maintained uniformly throughout certain restricted domains of the complex z-plane. 1) must obviously remain finite at such a point. J. solutions, after Wentzel  (1926), Kramers  (1926), Brillouin  (1926) and Jeffreys  (1923).
J. solutions are valid only in certain well-defined sectors around z = 0. 5) can be made equivalent to the comparison equation by choosing ξ ≡ j = − h2 qz′2, so ξ 1/2 dξ = ±ihq 1/2dz, or where ξ vanishes when q vanishes. 5) is neglected, then X˝ = ξX, possessing the two standard solutions Ai (ξ) and Bi (ξ), being the Airy integral and its companion function which are studied in Chapter III and in the Appendix. This yields It will be seen later that this approximate solution is not only valid in a domain completely surrounding z = 0, but also at z = 0 itself.
Langer  (1934), in a published lecture, complained that Jeffreys' method merely patched ‘the results together into a more or less complete representation’. He therefore attacked the problem from a more fundamental standpoint, namely ‘that of obtaining at one and the same stroke a representation which is valid over the whole domain of the variable’. J. 7). Wasow  (1960) has recently discussed the difference between Jeffreys' and Langer's approach. 5). Applications to physical problems have been less varied, since in the calculation of eigenvalues and reflection coefficients the value of the approximate solution at the transition point is not required.
An Introduction to Phase-Integral Methods by John Heading