John Heading - An Introduction to Phase-Integral Methods

A graph y = g(x) ef (x, h) > 0 is given for x > 0, such that when h is large ef attains large values for a restricted range R of values of x, while for other values of x ef is small; g(x) varies but slowly throughout R. The full line in represents the function considered. We seek an approximate value of the integral the convergence of the integral being assumed. FIG. 35 We replace the hump (usually formed by a non-integrable function) by a similar hump (formed by an integrable function), the two curves differing only appreciably for values of x outside R.

If the maximum value of (x) occurs at x = x0, the first two non-vanishing terms of the Taylor-series expansion about the point x = x0 are where (x0) is negative for a maximum. If xx0 = s, and if g(x) is replaced by g(x0) throughout R, we have approximately where the limits are used in order to facilitate the integration. In the figure, the dotted line represents the curve under which we are calculating the area. If , we have using the standard infinite integral exp ( u2) du = . This technique is a simplification of the method known as the method of steepest descents in the complex plane.

If h is real and positive, (h) is defined by the value of the area The graph of the integrand when h is large consists of a tall narrow peak. If h is real and positive, (h) is defined by the value of the area The graph of the integrand when h is large consists of a tall narrow peak.

An integration by parts shows immediately that (h + 1) = h(h), so , we have f(x) = x + h log x, f(x) = 1 + h/ x, f(x) = h/ x2. f(x) vanishes when x = h, so f(h) = h + h log h and f(h) = 1/ h; when h is large, the third derivative would be negligible. Equation () now gives This approximation is known as Stirling's formula. If h is complex, such that < arg h < , a more complete defini-tion of (h) is required, but the same approximate formula is valid, provided arg h lies within this restricted range.

The differential equation w = zw possesses two simple power-series solutions, single-valued and convergent for all values of z. 4z6/6! + 1.4. 7z9/9! + ..., w2 = z + 2z4/4! + 2.5z7/7! + 2.5.8zl0/10! + ... = z2 say. = z2 say.

The general solution is given by w = A w1 + Bw2. For large values of z, these series must be transformed into other forms in order to exhibit their true nature. When z = h (real, large and positive) each term in w1 is positive, and its approximate sum, following Stokes [], may be found by replacing the discrete terms in the series by a continuous variable. The term involving h3n is We therefore consider the continuous graph the area under the graph for positive x being the approximate sum of the series w1. The individual terms in w1 (and hence the value of y) . When x is large, the use of formula () yields where x2/3 is a slowly varying function contrasted to the exponent of the exponential.

If f = 2 x log x 2 x log 3 + 3 x log h + 2 x,

then	f = 2 log x 2 2 log 3 + 3 log h + 2 x,

f = 2/ x. Now f vanishes when at which pointand f = 6 h3/2. Hence, using formula (), we obtain a dominant expansion. This is one of the W.K.B.J. solutions of the Airy equation, but with a definite numerical coefficient. Similarly, w2 is replaced by where again h is real, large and positive.

As before, the maximum of the integrand occurs whenyielding Two standard tabulated solutions [] are taken to be for reasons now to be discussed. ) must be subdominant along the positive real axis; this method does not however produce the subdominant expression. Similarly, substitution into () yields after simplification