By John Heading
Since the variety of purposes is great, the textual content considers just a short collection of themes and emphasizes the strategy itself instead of specific functions. the method, as soon as derived, is proven to be one among crucial simplicity that comprises purely the appliance of convinced well-defined ideas. beginning with a ancient survey of the matter and its options, matters 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 resources for An Introduction to Phase-Integral Methods
In the z-plane, these lines are given by the anti-Stokes lines three of which emerge from O, with OA′ and OB′ coinciding. The branch cut is usually placed along this line. The solution e− ξ is uniformly valid all round z = 0 except on the line A′ OB′ and of course near O itself. This theory, following and extending that given by Jeffreys , does not demonstrate how far outwards these uniformly asymptotic solutions may be extended. Further zeros of q complicate the solution, FIG. 2. ξ-plane z-plane but a detailed examination of this problem has been given by Heading .
The use of approximations in mathematical physics is so familiar that it is surprising that some have complained about the point of view adopted in phase-integral methods. For example, with no just foundation for such remarks, Smyth  has criticized a paper making use of the method by writing: ‘It should be observed that the authors have used a solution which is a very poor approximation to the given problem as an approximate solution to another problem. ’ Concerning approximations, Schelkunoff  has more wisely remarked that there is ‘something in human nature that makes one yearn for the exact answer to a given problem.
Several texts have followed B. Jeffreys' treatment; for example, Mott and Sneddon , Wave Mechanics and its Applications. 9 Applications of the method Hartree  (1931) was the first investigator to apply these techniques to an isotropic loss-free ionosphere, considering two models with a linear and parabolic distribution of free electron density respectively. He solved the two problems firstly by ray methods, thinking in terms of the standard optical approach; then he solved the two problems by exact analytical methods, finding the asymptotic form of his answer for higher frequencies.
An Introduction to Phase-Integral Methods by John Heading