1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

86 CHAPTER 2 • COMPLEX FUNCTIONS

We can extend the system of complex numbers by joining to it an "ideal"
point denoted by oo and called the point at infinity. This new set is called the
extended complex plane. You will see shortly that the point oo has the property,

loosely speaking, that lim z = oo iff lim lzl = oo.

n-oo n-+oo

An c neighborhood of the point at infinity is the set { z : I zl > ~}. The usual

way to visualize the point at infinity is by using what we call the stereographic

projection, which is attributed to Riemann. Let n be a sphere of diameter 1 that

is centered at the point (0, 0, D in three-dimensional space where coordinates

are specified by the triple of real numbers (x, y, {). Here the complex number

z = x + iy is associated with t he point z = (x, y, 0).

The point N = (0, 0, 1) on n is called the north pole of n. If we let z be

a complex number and consider the line segment L in three-dimensional space

that joins z to the north pole N = (0, 0, 1), then L intersects fl in exactly one

point z. T he correspondence z <-> Z is called the stereographic projection of the

complex z plane onto the Riemann sphere n.

A point z = x + iy = (x, y, 0) of unit modulus will correspond with Z =

( ~, ~, !). If z has modulus greater than 1, then Z will lie in the upper hemisphere

where for points Z = (x, y, {) we have { > !· If z has modulus less than 1,

then Z will lie in the lower hemisphere where for points Z = (x, y, {) we have

{ < ~. The complex number z = 0 = 0 + Oi corresponds with the south pole,

§ = (0, 0, O). Now you can see that indeed z ~ oo iff lzl ~ oo iff Z ~ N.

Hence N corresponds with the "ideal" point at infinity. The situation is shown

in Figure 2.22.

Let's reconsider the mapping w = ~ by assigning t he images w = oo and

w = 0 to the points z = 0 and z = oo, respectively. We now write the reciprocal

transformation as

w = f(z) = { t

when z f: 0 and z f: oo;

when z = oo;

when z = 0.

(2-32)

Note that the transformation w = f (z) is a one-to-one mapping of the

extended complex z plane onto the extended complex w plane. Further, f is a

Figure 2. 22 The Riemann sphere.