25.8 APPROXIMATIONS TO INTEGRALS
are self-cancelling, as discussed previously. However, the ends of the contourmust
be in regions where the integrand is vanishingly small, and so at each end of
the level line we need to add a further section of path that makes the contour
terminate correctly.
Fortunately, this can be done without adding to the value of the integral. Thisis because, as noted in the second paragraph following equation (25.67), far from
the saddle the level line will be at a finite height up a ‘precipitous cliff’ that
separates the region where the integrand grows without limit from the one where
it tends to zero. To move down the cliff-face into the zero-level valley requires
an ever smaller step the further we move away from the saddle; as the integrand
is finite, the contribution to the integral is vanishingly small. In figure 25.12, this
additional piece of path length might, for example, correspond to the infinitesimal
move from a point on the large positivex-axis (whereh(z) has value 1) to a point
just above it (whereh(z)≈0).
Now that formula (25.73) has been justified, we may note that it is exactlythe same as that for the method of steepest descents, equation (25.66). A similar
calculation using the level line on which the phase is a maximum also reproduces
the steepest-descents formula. It would appear that ‘all roads lead to Rome’.
However, as we explain later, some roads are more difficult than others. Where
a problem involves using more than one saddle point, if the steepest-descents
approach is tractable, it will usually be the more straight forward to apply.
Typical amplitude-phase diagrams for an integration along a level line thatgoes straight through the saddle are shown in parts (a) and (b) of figure 25.14.
The value of the integral is given, in both magnitude and phase, by the vector
vjoining the initial to the final winding points and, of course, is the same in
both cases. Part (a) corresponds to the case of the phase being a minimum at the
saddle; the vector path crossesvat an angle of−π/4. When a path on which the
phase at the saddle is a maximum is used, the Cornu spiral is as in part (b) of
the figure; then the vector path crossesvat an angle of +π/4. As can be seen,
the two spirals are mirror images of each other.
Clearly a straight-through level line path will start and end in different zero-level valleys. For one that turns through a right angle at the saddle point, the
end-point could be in a different valley (for a function such as exp(−z^2 ), there is
only one other) or in the same one. In the latter case the integral will give a zero
value, unless a singularity ofh(z) happens to have been enclosed by the contour.
Parts (c) and (d) of figure 25.14 illustrate the phase–amplitude diagrams for these
two cases. In (c) the path turns through a right angle (+π/2, as it happens) at the
saddle point, but finishes up in a different valley from that in which it started. In
(d) it also turns through a right angle but returns to the same valley, albeit close
to the other precipice from that near its starting point. This makes no difference
and the result is zero, the two half spirals in the diagram producing resultants
that cancel.