1550251515-Classical_Complex_Analysis__Gonzalez_

60 Chapter^1

To determine analytically the rectangular coordinates (a, /3, 'Y) of
the point Q corresponding under stereographic projection to the point
P(x, y, 0) representing z = x + iy, we have

x y -1

(1.17-2)

since the three points P, Q, and N lie on the same line. From (1.17-2)
it follows that

Hence

so that

x=--a /3

1 -"(

and y=--

1-"(

. a+ i/3

z=x+iy= --

1 -"(

_. a - i/3

z=x-iy= --

1-'Y

(1.17-3)

(1.17-4)

(1.17-5)

2 - a2+132

lzl = zz = ( ) 2 (1.17-6)

1-'Y

Since the point Q(a,/3,'Y) is on the sphere we have, by (1.17-1),

a^2 +f3

2

+'Y^2 -"(=0 (1.17-7)

Hence (J..17-6) becomes

2

lzl2 = 'Y - 'Y = 'Y

(1-"()^2 1-'Y

and it follows that

lzl2

'Y = 1 + lzl2'^1 - 'Y =^1 + lzl2

1

(1.17-8)

Then equations (1.17-2) can be written in the form

a /3 1

-=-=---

x y 1 + lzl2

and we obtain

x Rez

a= ---= ---

l+lzl2 l+lzl2

y Imz

/3 = 1 + lzl2 1 + lzl2

(1.17-9)