1550251515-Classical_Complex_Analysis__Gonzalez_

Elementary Functions 259

where A = a1 + ia2, B = a1 + ia 2 , C = 2(c 1 + ic 2 ), N = 2c 3. Then

we find that

1

a1 + a1 a1 -a1 C1

D=- a2 +a2 a2 -a2 C2

4i

M+M M-M C3

1

a1 C111 C1

= 2i a2 lll2 C2

M M C3

1

a1 + ia2 a1 + ia2 c1 + ic2

= 4 ii1 - iii2 C111 - ZC\12 C1 - ic2

M M C3

A B c

1

= - f3 A () = ~E

8

M M N

8

Thus D # 0 implies that E # O, and conversely.
A generalized bilinear transformation may have either no fixed point or
one, two, three, four, or infinitely many fixed points lying on a generalized
circle. The analysis of the different cases is somewhat involved, and we
must refer the reader for this and other details to J. A. Cabri [4].

Exercises 5.2

1. Let S be the set of all conjugate bilinear transformations. An iden-
   tity in Sis a transformation V E S such that (VVU)(z) = U(z) and
   (UVV)(z) = U(z). Show that any conjugate transformation of the form

V(z) = ~z_+ i~

Z"'(Z +a

with a complex, (3 and 1 reals, and aii + (31 = 1 is an identity in S.

2. Find the fixed points of the transformation w = az + b (a # 0).


3. Show that the inverse of a nonsingular bilinear transformation of the
   form (5.13-3) is given by

(AN - CM)w +(CM - BN)w +(BC - AC)

z = - --

(BM - AM)w +(BM - AM)w +(AA - BB)

4. Find the fixed points of w = Az + Bz + C, IAI # IBI # 0.