By John Heading
Since the variety of functions is significant, the textual content considers just a short choice of subject matters and emphasizes the tactic itself instead of distinctive functions. the method, as soon as derived, is proven to be considered one of crucial simplicity that comprises in basic terms the appliance of sure well-defined ideas. beginning with a ancient survey of the matter and its suggestions, topics comprise 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
J. J. solutions originates from an appropriate change of the dependent variable. The given equation is where the parameter h is taken to be large and positive, and where the function q(z, h) tends to a limit as h → ∞ for fixed complex z. A prime denotes differentiation with respect to z. If we let with a constant lower limit of integration, then w′ = ø exp (∫ ødz) and w˝ = ø2 exp (∫ ødz) + ø′ exp(∫ ødz), yielding ø2 + ø′ + h2 q = 0. , where each øj(z, h) tends to a limit as h → ∞ for fixed α, then substitution yields as far as terms in h2, h, h0 are concerned.
Since many problems that arise in theoretical physics lead to differential equations of this type, attempts must be made to produce satisfactory methods of solution. If p and q are specially chosen, the various standard transcendental differential equations are produced. For example, if p = 1/ x and q = 1 − v2/ x2, we obtain Bessel's equation of order v; if p = 0 and q = − x, we obtain the Airy equation. These special cases are singled out because exact solutions are readily obtainable in terms of contour integrals, from which the power-series solutions in ascending powers of x and the asymptotic solutions in descending powers of large x may be obtained.
Solution. In other words, we have traced the solution into a domain in which the original approximate representation is no longer valid. Boundaries across which such a change takes place are known as anti-Stokes lines; within such boundaries, the solution is uniformly asymptotic. J. 11). However, he did not elucidate the question of the boundaries in a systematic way. Olver  (1959) examined the equation with two transition points whose exact solutions are expressed in terms of Weber functions.
An Introduction to Phase-Integral Methods by John Heading
Categories: Differential Equations