Elementary Functions25. If a is an arbitrary point such that !al '/' R, then the function
F(z) = R
2
lal z -a
a az -R^2251has on the circle lzl =Ra constant absolute value equal to R, and so
transforms this circle into itself.- Find the most general bilinear transformation having i and -i as fixed
points.
5.12 The Conjugate Bilinear Function
The simplest function that defines a one-to-one mapping between C and
IC which is inversely isogonal (i.e., preserves angles but reverses their ori-
entation) is the conjugation function J(z) = z. As already noted, this
function defines geometrically a symmetry with respect to the real axis.
Composition of J(z) with a bilinear function w = T(z) yields the most
general one-to-one orientation reversing mapping, namely,
az+ b
w = TJ(z) = -_-= U(z)
cz+d
(5.12-1)which is called the conjugate bilinear function or the conjugate bilinear
transformation. We shall also normalize (5.12-1) by assuming that ad -
be = 1.
Transformations of this type do not form a group, since for any two such
transformations U 1 and U 2 , their composition product U1 U2 is bilinear, so
it is not of the form (5.12-1). However, for any three transformations U 1 ,
U 2 , U 3 we have (U 1 U2Ua) ES, where Sis the set of all transformations of
the form (5.12-1). In addition, the associative law
((U1U2Ua)U4Us) = (Ui(U2UaU4)Us) = (U1U2(UaU4Us))holds, as it is easy (but tedious) to verify. Hence the set S of the conjugate
bilinear transformations constitutes what is called in algebra a ternary
semigroup of transformations; see [3], [9], and [15].
Let z1, z2, za, Z4 be four distinct points, one of which may be oo, and
let Wk = U(zk)· If we set
then
(wi,w2,wa,w4) = X
so the cross ratio of four points is not invariant under a conjugate bilinear
transformation, except when ,\ = X, i.e., when ,\ is real, in which case