By John Heading
Since the variety of purposes is big, the textual content considers just a short number of issues and emphasizes the tactic itself instead of certain purposes. the method, as soon as derived, is proven to be one in all crucial simplicity that comprises only the appliance of sure well-defined ideas. beginning with a old survey of the matter and its strategies, 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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Extra info for An Introduction to Phase-Integral Methods
Connection formulae provide the answer to this question. A substantial metamorphosis evidently takes place in the form of the function, under which sines and cosines must smoothly join on to exponential functions. This problem is treated with various degrees of rigour and clarity in several texts. For example, the reader may consult Jeffreys and Jeffreys , Methods of Mathematical Physics, or Morse and Fesh-bach , Methods of Theoretical Physics, or various books on wave mechanics, such as Kemble , The Fundamental Principles of Quantum Mechanics, or Bohm , Quantum Theory.
The Stokes phenomenon and the Airy integral is discussed in Chapter III, while an account based on contour integration may be found in Budden's text , Chapter XV. Briefly, the reason why a discontinuity is allowed in a solution that is strictly continuous is because asymptotic series always contain an inherent error, and the discontinuity that occurs is less than this inherent error. An asymptotic series is really divergent, but nevertheless it can be summed with great accuracy for large |z| by employing only those terms for which the magnitude of the terms decreases along the series.
B. Jeffreys  (1942) used Jeffreys' connection formulae across the two transition points, obtaining the correct answer to a certain degree of accuracy. 4) from right to left, and the modified formula, as we have already pointed out, is the only one that yields the correct answer. It appears, then, that it was a coincidence that the correct answer was obtained, since no explanation had then been given why that formula and none other was essential in such problems. Rydbeck  (1948) also gave an independent examination of the same problem.
An Introduction to Phase-Integral Methods by John Heading