Mathematical Methods for Physics and Engineering : A Comprehensive Guide

25.6 STOKES’ EQUATION AND AIRY INTEGRALS the integrand. Consider the contour integral y(z)= ∫b a f(t) exp(zt)dt, (25.33) in whic ...

APPLICATIONS OF COMPLEX VARIABLES C 1 C 2 C 3 Ret Imt Figure 25.10 The contours used in the complext-plane to define the functio ...

25.6 STOKES’ EQUATION AND AIRY INTEGRALS contour integral (25.35) with the particular solution of Stokes’ equation that decays m ...

APPLICATIONS OF COMPLEX VARIABLES z − 10 − 5 − 0. 5 0. 5 1 Figure 25.11 The functions Ai(z) (full line) and Bi(z) (broken line) ...

25.7 WKB methods there exist many representations, both as linear combinations and as indefinite integrals, of one in terms of t ...

APPLICATIONS OF COMPLEX VARIABLES x-dependence ofn(x), the solution would be (as usual) y(x)=Aexp(−ik 0 nx), (25.43) with bothAa ...

25.7 WKB METHODS This still requiresk 0 n′(x) to be small (compared with, say,k^20 n^2 (x)), but is some improvement (not least ...

APPLICATIONS OF COMPLEX VARIABLES which there is no obvious solution for a generaln(x). However, on the assumption thatd^2 φ/dx^ ...

25.7 WKB METHODS The precise combination of these two solutions that is required for any particular problem has to be determined ...

APPLICATIONS OF COMPLEX VARIABLES
Apply the WKB method to the problem of finding the quantum energy levelsEof a particle of mas ...

25.7 WKB METHODS for some constantA. This form is only valid for negativezclose toz= 0 and is not appropriate within the well as ...

APPLICATIONS OF COMPLEX VARIABLES The result just proved gives ∫an −an √ 2 mV 0
(a^2 ns−x^2 s)^1 /^2 dx=(n+^12 )π. Writingx=va ...

25.7 WKB METHODS one function and the same function may need different expansions for different values of argz. Finally in this ...

APPLICATIONS OF COMPLEX VARIABLES exp[±iφ(z) ] will be purely real. What is more, one such factor, known as the dominantterm, wi ...

25.8 Approximations to integrals Finally, we should mention that the lines in thez-plane on which the expo- nents in the WKB sol ...

APPLICATIONS OF COMPLEX VARIABLES Figure 25.12 A greyscale plot with associated contours of the value of|h(z)|, whereh(z)=exp[i( ...

25.8 APPROXIMATIONS TO INTEGRALS From the observations contained in the two previous paragraphs, we deduce that a path that foll ...

APPLICATIONS OF COMPLEX VARIABLES angles) in which the phase ofh(z) is independent ofρ. They make angles ofπ/ 4 with the level l ...

25.8 APPROXIMATIONS TO INTEGRALS any of the continuations to infinity of the level lines that pass through the saddle. In practi ...

APPLICATIONS OF COMPLEX VARIABLES sign is determined by the directionθin the complex plane in which the distorted contour passes ...