Mathematical Methods for Physics and Engineering - Matematica.NET

APPLICATIONS OF COMPLEX VARIABLESvv(a)(b)v(c)(d)Figure 25.14 Amplitude–phase diagrams for stationary phase integration. (a)Using a straight-through path on which the phase is a minimum. (b) Usinga straight-through path on which the phase is a maximum. (c) Using a levelline that turns through +π/2 at the saddle point but starts and finishes indifferent valleys. (d) Using a level line that turns through a right angle butfinishes in the same valley as it started. In cases (a), (b) and (c) the integralvalue is represented by v (see text). In case (d) the integral has value zero.We do not have the space to consider cases with two or more saddle points,but even more care is needed with the stationary phase approach than whenusing the steepest-descents method. At a saddle point there is only one l.s.d. butthere are two level lines. If more than one saddle point is required to reach theappropriate end-point of an integration, or an intermediate zero-level valley hasto be used, then care is needed in linking the corresponding level lines in such away that the links do not make a significant, but unknown, contribution to theintegral. Yet more complications can arise if a level line through one saddle pointcrosses a line of steepest ascent through a second saddle.We conclude this section with a worked example that has direct links to thetwo preceding sections of this chapter.918