142 4. Equations
v(r cos 0, r sin 0) = r” sin n0 + terms of lower degree in r.
Since r is very large, the terms involving the highest powers of P in
the expansions of u and v will be dominant, and the terms of lower
degree will be negligible by comparison except where cos n0 and sin nr9
are near zero. Consequently, as 0 increases from 0 to 27r, it is to be
expected that u(r cos 6, r sin 0) changes sign roughly as r’” cos n0 and
v(r cos 0, r sin 0) changes sign roughly as r” sin n0.
Sketch the graphs of cosnt9 and sin nr9 as functions of 0 over the
domain 0 5 0 < 2n, and verify that each has 2n zeros, those of one
function interlacing those of the other.
If we mark with an R the intersection of the largest circle and the
locus of U(X, y) = 0 and with an I the intersection of the large circle
and the locus of v(x, y) = 0, argue that the points R and I will be
approximately evenly spaced around the circumference of the circle
and the R-points will alternate with the I-points.
The final step is to argue that the part of the locus of u(x, y) = 0
inside the circle consists of a number of curves connecting pairs of
R-points and the part of the locus of v(x, y) = 0 inside the circle
consists of a number of curves connecting pairs of I-points, and then
to deduce that the two loci must inevitably intersect. Check this in the
case of the quintic: indicate on a circle ten points R and ten points
I alternating with the R-points; join pairs of points with the same
letter and check the plausibility that some line joining 2 R-points
must intersect some line joining 2 I-points.4.5 The Fundamental Theorem: Functions of a
Complex Variable
A second approach to the Fundamental Theorem involves the idea of curves
winding around the origin. To handle this, we need some way of visualiz-
ing the action of functions of complex variables. In the case of real func-
tions, this is done by sketching graphs in the plane. However, since the
space of complex numbers has two real dimensions, we would need a four-
dimensional space in which to construct the graph of a complex function
of a complex variable. We avoid this by looking at two complex planes,
one for the domain of the function and the other for the range. Suppose
h(z) is a function of the complex variable z and w = h(z). For each z in
the complex plane of the domain (the z-plane), we plot the corresponding
point w = h(r) in the plane of the range (the w-plane). We write z = x+yi
and w = u + vi.
This in itself is not very useful. To get a sense of how the function h
behaves, it is better to envisage z as a moving point in the z-plane and