1550251515-Classical_Complex_Analysis__Gonzalez_

Elementary Functions

we have

1

M+ M >2

and it follows from (5.7-5) that a+ d must be real, and

la+dl >2

231

(5.8-3)

That this condition is also sufficient, thus characterizing the hyperbolic
transformations, will be seen. shortly.

(b) M = eilJ (R = 1, () "=f. 2h). In this subcase the multiplier is unimod-

ular. Equation (5.8-2) is now of the form W = eilJ Z, and the transformation
is a pure rotation about the origin, so it has the following properties:

1. Each straight line passing through the origin is carried into another
   straight line through the origin that makes an angle () with the first.


2. Each circle about the origin is transformed into itself. The interior and
   exterior of each circle are mapped into themselves. The points 0 and
   oo are fixed and are inverses of each other with respect to any one of
   those circles (Fig. 5.7).

By applying the transformations s-^1 , so as to return to the original
variables, the points 0 and oo are carried into the fixed points z1 and z2,
respectively, and we have:

1. Every arc z]Z2 of a circle passing through the fixed points is trans-
   formed into another arc of a circle with the same endpoints, the second
   arc making an angle () with the first.


2. Every circle orthogonal to those passing through the fixed points is
   mapped into itself. The interior and exterior of each such circle are

Fig. 5.7