1550251515-Classical_Complex_Analysis__Gonzalez_

234 Chapter^5

where M = a^2 • The substitutions Z = z - z 1 , W = w - w 1 brings it
to the form

W=MZ
so (5.8-6) represents a homotecy with center z 1 if a is real, a rotation about
z 1 ifa = efo(a -:j; k7r), or a combination of both ifa = Aei<>(A # 1, a# k7r).
The transformation could be also called hyperbolic, elliptic, or loxodromic,
respectively, since a + d in those cases satisfies the same conditions as in
cases (a), (b), and (c) discussed above.
( e) F'inally, we consider the case where the transformation has a double

fixed point. This case arises when H = 0, i.e., when a+ d = ±2. According

to Klein, the transformation of this type are called parabolic.

If c = O, b # 0, then ad = 1 and, in view of a + d = ±2, we have

a= d = ±1. Hence the transformation becomes the translation

w=z±b (5.8-7)

which has oo as a double fixed point.

If c -:f:. 0, the double fixed point is z 1 = ( a-d)/2c, and the transformation

reduces to the form (5.8-7), in fact, to the translation

W=Z±c

(5.8-8)

by the change of variables

Z=

1

W=

1
(5.8-9)
Z-Z1 w-z 1
as follows from (5. 7-7).
The transformation (5.S-8) carries every straight line into a parallel line,
except when the line is parallel to the support of the vector c. In that case
the line is mapped into itself; also, each half-plane determined by such a
line is mapped into itself.
Returning to the original variables by means of (5.8-9), the point oo goes
into z1, and considering that each of (5.8-9) represents an inversion with
center z 1 followed by a symmetry, we conclude the following:

1. Each circle passing through the fixed point z 1 is carried into another
   circle through z1 which does not intersect the first, that is, into a
   tangent circle to the first at z 1. '


2. There is a family of circles tangent at z 1 with the property that each
   circle of the family is carried into itself. Also, the interior (or exterior)
   of each of those circles is carried into itself.
   Figure 5.9 represents two families of orthogonal lines, namely, a family
   of lines parallel to Oc, and the other a family of lines perpendicular to