APPLICATIONS OF COMPLEX VARIABLES
Figure 25.12 A greyscale plot with associated contours of the value of|h(z)|,
whereh(z)=exp[i(z^3 +6z^2 − 15 z+ 8)], in the neighbourhood of one of its
saddle points; darker shading corresponds to larger magnitudes. The plot also
shows the two level lines (thick solid lines) through the saddle and part of the
line of steepest descents (dashed line) passing over it. At the saddle point, the
angle between the line of steepest descents and a level line isπ/4.level linesof the function. It follows that the direction in which the magnitude
ofh(z) changes most rapidly at any pointzis in a direction perpendicular to
the level line passing through that point. This is therefore the line throughzon
which the phase ofh(z), namelyψ(z), is constant. Lines of constant phase are
therefore sometimes referred to aslines of steepest descent(or steepest ascent).
We further note that|h(z)|can never be negative and that neitherφnorψcan have a finite maximum at any point at whichf(z) is analytic. This latter
observation follows from the fact that at a maximum of, say,φ(x, y), both∂^2 φ/∂x^2
and∂^2 φ/∂y^2 would have to be negative; if this were so, Laplace’s equation could
not be satisfied, leading to a contradiction. A similar argument shows that a
minimum of eitherφorψis not possible whereverf(z) is analytic. A more
positive conclusion is that, since the two unmixed second partial derivatives
∂^2 φ/∂x^2 and∂^2 φ/∂y^2 must have opposite signs, the only possible conclusion
about a point at which∇φis defined and equal to zero is that the point is a
saddle point ofh(z). An example of a saddle point is shown as a greyscale plot
in figure 25.12 and, more pictorially, in figure 5.2.