An Introduction to Phase-Integral Methods by John Heading

Author:John Heading
Language: eng
Format: epub
Publisher: Dover Publications
Published: 2013-06-08T16:00:00+00:00

In Chapter II, we have seen that differential equations of the form

w˝ + h2q(z, h)w = 0

often admit of approximate W.K.B.J. solutions in the form

in restricted domains of the complex z-plane.

We now consider how more complete solutions may be obtained, solutions that do not suffer from this restriction on their domains of validity. To this end, we consider the simplest yet most usual case of one transition point of order unity. That is to say, q possesses a zero at the point z = z0 (chosen to be zero for convenience) such that in its neighbourhood q exists as a convergent power series of the form

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