Mathematical Methods for Physics and Engineering - Matematica.NET

25.7 WKB METHODSone function and the same function may need different expansions fordifferent values of arg z.Finally in this subsection we note that, although the form of equation (25.42) mayappear rather restrictive, in that it contains no term in y ′ , the results obtained sofar can be applied to an equation such asd 2 ydz 2 + P (z)dy + Q(z)y =0. (25.56)dzTo make this possible, a change of either the dependent or the independentvariable is made. For the former we write( ∫ 1 zY (z) =y(z)exp P (u) du)⇒d2 Y(Q2dz 2 + − 1 4 P 2 − 1 )dPY =0,2 dzwhilst for the latter we introduce a new independent variable ζ defined by( ∫dζz)( ) 2dz =exp − P (u) du ⇒d2 y dzdζ 2 + Q y =0.dζIn either case, equation (25.56) is reduced to the form of (25.42), though itwill be clear that the two sets of WKB solutions (which are, of course, onlyapproximations) will not be the same.25.7.4 The Stokes phenomenonAs we saw in subsection 25.7.2, the combination of WKB solutions of a differentialequation required to reproduce the asymptotic form of the accurate solution y(z)of the same equation, varies according to the region of the z-plane in which z lies.We now consider this behaviour, known as the Stokes phenomenon, in a littlemore detail.Let y 1 (z) andy 2 (z) be the two WKB solutions of a second-order differentialequation. Then any solution Y (z) of the same equation can be written asymptoticallyasY (z) ∼ A 1 y 1 (z)+A 2 y 2 (z), (25.57)where, although we will be considering (abrupt) changes in them, we will continueto refer to A 1 and A 2 as constants, as they are within any one region. In order toproduce the required change in the linear combination, as we pass over a Stokesline from one region of the z-plane to another, one of the constants must change(relative to the other) as the border between the regions is crossed.At first sight, this may seem impossible without causing a discernible discontinuityin the representation of Y (z). However, we must recall that the WKBsolutions are approximations, and that, as they contain a phase integral, forcertain values of arg z the phase φ(z) will be purely imaginary and the factors903