An Introduction to Phase-Integral Methods by John Heading

An Introduction to Phase-Integral Methods by John Heading

By John Heading

The phase-integral approach in arithmetic, sometimes called the Wentzel-Kramers-Brillouin (WKB) technique, is the point of interest of this introductory remedy. writer John Heading effectively steers a direction among simplistic and rigorous ways to supply a concise evaluate for complex undergraduates and graduate scholars in arithmetic and physics.
Since the variety of purposes is sizeable, the textual content considers just a short choice of themes and emphasizes the tactic itself instead of precise purposes. the method, as soon as derived, is proven to be considered one of crucial simplicity that includes simply 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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For example, with no just foundation for such remarks, Smyth [102] 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 [97] has more wisely remarked that there is ‘something in human nature that makes one yearn for the exact answer to a given problem. J. method’. Phase-integral methods are applied to the particular equation in which h is a large parameter and q(x, h) a variable function of x and h.

The integral equation, soluble by successive substitutions, is derived by the same technique as differential equations are solved by the method of variation of parameters. If Z1(ξ) and Z2(ξ) are two solutions of the comparison equation X˝ − jX = 0, let us seek to express X(ξ) in the form X(ξ) = α(ξ) Z1(ξ) + β(ξ) Z2(ξ). Since two unknown functions α(ξ) and β(ξ) are involved, and only one equation must be satisfied, any useful relationship may be postulated between α(ξ) and β(ξ). It follows upon differentiation that X ′ =α ′ Z1 +α Z ′1 +β ′ Z2 +α Z ′2.

Jeffreys [63] (1942), for example, took this point of view in a potential barrier problem. 4) she would have obtained a different answer. 4), this suggestion rightly indicating that the left-hand side does imply the right-hand side. He goes on to say that if such were so, no information could be deduced in problems with boundary conditions on both sides of the origin, and ‘in some cases results have had an accuracy far greater than there was any apparent reason to expect’. Jeffreys then considers two problems.

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