Mathematical Methods for Physics and Engineering : A Comprehensive Guide

APPLICATIONS OF COMPLEX VARIABLES

(a) (b)

(c) (d)

v v

v

Figure 25.14 Amplitude–phase diagrams for stationary phase integration. (a)

Using a straight-through path on which the phase is a minimum. (b) Using

a straight-through path on which the phase is a maximum. (c) Using a level

line that turns through +π/2 at the saddle point but starts and finishes in

different valleys. (d) Using a level line that turns through a right angle but

finishes in the same valley as it started. In cases (a), (b) and (c) the integral

value is represented byv(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 when

using the steepest-descents method. At a saddle point there is only one l.s.d. but

there are two level lines. If more than one saddle point is required to reach the

appropriate end-point of an integration, or an intermediate zero-level valley has

to be used, then care is needed in linking the corresponding level lines in such a

way that the links do not make a significant, but unknown, contribution to the

integral. Yet more complications can arise if a level line through one saddle point

crosses a line of steepest ascent through a second saddle.

We conclude this section with a worked example that has direct links to the

two preceding sections of this chapter.