1550251515-Classical_Complex_Analysis__Gonzalez_

128 Chapter^3

To sidestep such inconvenience, a two-dimensional surface (usually, the
Gaussian plane or the Riemann sphere) is used to represent the variable
z, and another surface to represent the variable w (which may or may not
be superimposed on the first). Then the mapping that any given function
defines from one surface into the other is visualized by considering the

images under f of certain subsets of the domain of f, ordinarily some

simple geometric figures (points, straight lines, circles, conics, etc.) as well
as the regions bounded by them.
For example, under w = z^2 the line z = x (the x-axis) maps into w = x^2
(the nonnegative u-axis), and the line z = iy (the y-axis) maps into w =
-y^2 (the nonpositive u-axis). Also, the half-line z =a+ it (a> 0,t 2:: 0)
maps into the arc u = a^2 - t^2 , v = 2at (upper part of a parabola), while
the half-line z = t + ic ( c > 0, t 2:: 0) maps into the arc u = t^2 - c^2 , v = 2ct
(upper part of an orthogonal parabola), and the region bounded by the
rectangle OABCO maps into the region bounded by O'A'B'C'O', as can
be verified by testing some interior point, say (1/ 2 a,^1 / 2 c) (Fig. 3.1).
A circle lzl = r = const. maps, under the same function, into the circle
lwl = r^2 , but the circle z = 1 + eit (0 :St :S 27r) maps into the cardioid
w = 1 + 2eit + e^2 it (Fig. 3.2). Since z = 1 goes into w = 1, the disk
lz -l I < 1 maps into the interior of the cardioid.
In Chapter 5 we discuss in some detail the geometry of the mapping
defined by a number of other particular functions.
Other proposed geometric representations are the following:

1. According to Briot and Bouquet, u = u( x, y) and v = v( x, y) can

be represented separately by surfaces, and the set of the two surfaces over

the domain D of definition of f is taken as a geometric description of the

behavior of f over D.

v

y

z-plane w-plane

c B

0 A x C' O' u

Fig. 3.1