1550251515-Classical_Complex_Analysis__Gonzalez_

Complex Numbers 61

Equations (1.17-9) together with (1.17-8) determine a, (3, 'Yin terms of z.

Conversely, equations (1.17-3) or, equivalently, (1.17-4) give z in terms of

the coordinates of Q.

Instead of determining Q by means of its Cartesian coordinates a, (3,

"( we may as well fix its position by means of the colatitude .A = LN GQ

(0 < .A :S 7r) measured from GN, where G is the center of the sphere, and

the azimuth()= LXOP (0 :S () < 27r) measured from OX. Clearly,t

Argz = Btan^1 M7r - .A)= lzl (1.17-10)

so that

() = Argz, A= 7r - 2tan-^1 Jzl (1.17-11)

It follows from the formulas above that the lines through the origin in

the complex plane Arg z = const. map into the meridians () = const. (i.e.,

great circles through 0 and N), and that'the circles lzl = r map into the

parallels of latitude .A = const. In particular, the unit circle lzl = 1 maps

into the equator A = 7r /2. This can also be seen from (1.17-8), which

reduces to 'Y =^1 / 2 for lzl = 1. The interior of the unit circle maps into

the southern hemisphere, while the exterior of the unit circle maps into

the northern hemisphere with the point N deleted. As the point z recedes

from the origin, i.e., as lzl --+ +oo, the corresponding point Q on the sphere

approaches the north pole N, since the second equation in (1.17-11) gives

.A --+ 0, or, alternatively, (1.17-8) gives 'Y --+ 1 as lzl --+ +oo. For this reason

the point N is said to correspond to the "point at infinity" of the plane

C, which explains the notation oo used at the beginning of this section to

denote the ideal or improper point of the complex plane.

It should be pointed out that the extended complex plane C* differs

from the so-called projective plane. The latter is obtained by adjoining

to the finite plane an infinite set of improper points (the line at infinity).

An image of the projective plane can be obtained by mapping the plane

C upon S from the center G of the sphere (Fig. 1.14). Then C maps into

the southern hemisphere and the equator is regarded as the image on S of

the line at infinity of the plane.

We also recall in connection with the real number system (the real line

in the geometric representation) that two ideal points -oo and +oo are

introduced. This is required by the linear ordering of the reals. Since no
such ordering exists for the complex numbers, as a matter of convenience,
just a single ideal point needs to be adjoined to the complex plane.

tHere we are choosing the azimuth as the principal value of the argument.