25.8 APPROXIMATIONS TO INTEGRALS
(a)(b)(c)βπ/ 4√
πFigure 25.13 Amplitude–phase diagrams for the integral∫∞
−∞exp(−z(^2) )dz
using different contours in the complexz-plane. (a) Using the real axis, as in
the steepest descents method. (b) Using the level linez=uexp(−^14 iπ)that
passes through the saddle point, as in the stationary phase method. (c) Using
a path that makes a positive angleβ(<π/4) with thez-axis.
The final curve, 25.13(c), shows the amplitude-phase diagram corresponding to
an integration path that is along a line making a positive angleβ(0<β<π/4)
with the realz-axis. In this case, the constituent infinitesimal vectors vary in both
length and direction. Note that the curve passes through its centre point with the
positive gradient tanβand that the directions of the spirals around the winding
points are reversed as compared with case (b).
It is important to recognise that, although the three paths illustrated (and the
infinity of other similar paths not illustrated) each produce a different phase–
amplitude diagram, the vectors joining the initial and final points in the diagrams
areall the same. For this particular integrand they are all (i) parallel to the
positive real axis, showing that the integral is real and giving its sign, and (ii) of
length
√
π, giving its magnitude.
What is apparent from figure 25.13(b), is that, because of the rapidly varying
phase at either end of the spiral, the contributions from the infinitesimal vectors
in those regions largely cancel each other. It is only in the central part of the
spiral where the individual contributions are all nearly in phase that a substantial
net contribution arises. If, on this part of the contour, where the phase is virtually