1550251515-Classical_Complex_Analysis__Gonzalez_

Elementary Functions 237

and

(x1,x2,xa,z) = (w1,w2,wa,w*)

Since (x1,x 2 ,x 3 ,z) = (x 1 ,x 2 ,x 3 ,z), equation (5.9-1) holds. Conversely,

if (5.9-1) holds and (x1, x 2 , xa, () = ( w 1 , w 2 , w 3 , w*), it follows at once that

( = z, and so w* is the symmetric of w with respect to C.

In the next theorem it is shown that symmetric points with respect to

a circle are inverses of each other with respect to the same circle.

Theorem 5. 7 If the points w and w* are symmetric with respect to a

circle C: IW -al = r, then

(w* - a)(w :__a)= r^2 (5.9-2)

Proof Suppose that w1, w2, wa are three distinct points on C. We have

( W1' W2' W3' w) = ( W1 - a, W2 - a, W3 - a, w -a)

= (w1 - a,w2 - a,wa - a,w - a)

= (__c_, _r_2_' __c_,w-a)

w1 - a w2 - a wa - a

(w1 - a, w2 - a, wa - a, _ r

2

_ )

w-a

r2

= ( Wi, W2, Wa, -=-----=-w-a +a)

which shows, by Theorem 5.6, that the symmetric of w is the point w* =

[r^2 /(w - a)]+ a. Hence wand w* satisfy (5.9-2).

Equation (5.9-2) yields lw - allw - al = r^2 ; i.e., the product of the
distances from the center a to the points w and w is r^2. Also, points w
and w* are both on the same ray from a, since

w* -a

w-a

r2

= --->0

lw-al2

Thus w and w* are indeed inverses of each other with respect to C.

Theorem 5.8 If a bilinear transformation T carries a circle C into a circle

r, then T maps a pair of symmetric points with respect to C into a pair of

symmetric points with respect to r (symmetry principle).

Proof If either C or r is the real axis, the property follows from the def-

inition of symmetry with respect to a circle. Otherwise, let w = Tz, and
U and V bilinear transformations such that x = Uz and x = Vw, where x
is real. Each of these transformations is determined by three pairs of cor-
responding points. Then T = v-^1 u, and since both U and v-^1 preserve
symmetry, so does T.