By John Heading
Since the variety of purposes is immense, the textual content considers just a short choice of subject matters and emphasizes the strategy itself instead of exact purposes. the method, as soon as derived, is proven to be one among crucial simplicity that comprises only the applying of sure well-defined ideas. beginning with a historic 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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Additional resources for An Introduction to Phase-Integral Methods
5) based on the radial wave equation in r as it stands, expecting a subdominant solution within the lower transition point to remain finite at the singularity r = 0, but such is not the case. The results they derived are, then, correct only when l is large, where l is related to the separation constant introduced when the r and θ equations are separated. Langer  (1937) has given the corect calculation, showing that a preliminary transformation is necessary to deal with the singularity at r = 0.
4) has been noted by Langer . Langer has also pointed out the danger of the use of the symbol ↔, showing that it can give rise to ‘misconceptions that have not failed to show themselves in the literature’. J. solution to be traced from the left to the right or from the right to the left, namely that the left implies the right and the right implies the left. This misconception arises because the definition of the symbol given by Jeffreys was not understood. J. solutions but not vice versa of necessity.
The method of variation of parameters was used once suitable approximate solutions for the left-hand side had been obtained. Following Langer, he introduced Hankel functions of order ⅓ rather than the standardized Airy functions, thereby in effect introducing singularities where strictly speaking none really can exist. J. solutions and Langer's solution in discussing the reception of radio signals from outer space by an aerial situated in the upper part of the ionosphere. Jeffreys  (1953) examined the representation of the solution through one transition point in terms of the Airy integral, but did not examine the boundary of the domain of validity in a systematic manner, and what happened at the transition point was not discussed.
An Introduction to Phase-Integral Methods by John Heading