1550251515-Classical_Complex_Analysis__Gonzalez_

236 Chapter 5

of the types called hyperbolic, elliptic, loxodromic, or parabolic, then the

binomial a+ d necessarily satisfies certain conditions. Since these condi-

tions exhaust all possibilities and are mutually exclusive, it follows, from a

general principle of logic, that those conditions are also sufficient. Hence,

summing up, we have the following theorem.

Theorem 5.5 Given a bilinear transformation w = ( az + b) / ( cz + d), not

the identical transformation, with ad - be = 1, the transformation is

hyperbolic iff a+ d is real and la+ di > 2,

elliptic iff a + d is real and la + di < 2,

parabolic iff a+ d is real and la + di = 2,

loxodromic iff a + d is nonreal.

Thus the type of bilinear transformation depends essentially on the trace

a + d of the matrix [:! ].

5.9 Symmetry with Respect to a Circle

A point z and its conjugate z = J(z) are symmetric (in the usual sense)

with respect to the real axis. If a bilinear transformation T maps the

real axis into a circle C, we regard the images of z and z by T, namely,

the points w = Tz and w* = Tz as symmetric (in a generalized sense)

with respect to the circle C. The mapping K that carries w into w*,

called a reflection with respect to C, is obviously one-to-one and such that

K = T JT-^1 • K is independent of T because if Sis some other bilinear

transformation mapping the real axis into C, then s-^1 T carries the reals
into the reals, and so has real coefficients. This implies that s-^1 T maps
a pair of conjugate points z, z into another pair of conjugate points, i.e.,
if ( = s-^1 Tz, then ( = s-^1 Tz, or s-^1 TJz = JS-^1 Tz, and it follows
that TJT-^1 = SJs-^1 •
In terms of the cross ratio, symmetry of two points with respect to a
circle can be characterized as follows:
Theorem 5.6 Given three distinct points w1, w 2 , w 3 on a circle C, the
points w and w are symmetric with respect to C iff
(5.9-1)
Proof Let T be a bilinear transformation mapping the real axis into C, and
let X1 = T-^1 wi, X2 = T-^1 w 2 , X3 = T-^1 w 3 , z = T-^1 w, z = T-^1 w, where
w* is the symmetric of w. By the invariance of the cross ratio we have

(xi,x2,x3,z) = (w1,w2,wa,w)