1550251515-Classical_Complex_Analysis__Gonzalez_

258 Chapter 5

By introducing, if necessary, an appropriate factor, we may assume the
conditions above to be

and

Hence general bilinear transformations having an inverse transformation
of the same type are of the form

w ==

Az+Bz+C

Mz+Mz+N

(5.13-3)

where N is real. Since the identity transformation is contained in (5.13-3)
for B == C == M = 0 and A = N, and the composition product of two
transformations of this type is again a transformation of the same form,
the set of all transformations of the particular form (5.13-3) constitutes a
group. In fact, it coincides with the group of all nonsingular collineations
of the projective plane, provided that the point at oo be replaced by an
improper line (the so-called line at infinity), and that

A B C

E= fJ A C #0

MM N

To check the last statement we recall that the collineations of the
projective plane are given by a pair of linear fractional equations

the collineations being nonsingular whenever

al bi C1

D=a2 bz c2#0

aa ba ca

(5.13-4)

Letting z = x + iy, w = u +iv, a1 = ai + ib1, °"2 = az + ib2, M = aa + iba,
and noticing that a 1 z + a 1 z = 2 Re( a1 z) = 2( a 1 x + b 1 y), and so on, we have

w= ii1z ~-'-~~~---'-+ a1z + 2c1+i(a2z+0t2z + 2c2)

=

Mz +Mz+ 2ca

Az+Bz+C

Mz+Mz+N