1550251515-Classical_Complex_Analysis__Gonzalez_

378 Chapter 6

angle is preserved under the transformation. For mappings between planes

a simple proof will be given by using the Kasner circle [47].

Theorem 6.27 Let f E 1J(N 0 (z)), and suppose that fz f O, fz f 0, and

J1(z) f 0. Then corresponding to any ray from a point z, with principal

argument 81 , there is an associated ray with the same initial point and

argument 82 such that the angle 82 - 81 is preserved in size as well as

orientation under f. In other terms, the nonconformal mapping defined by

f is nevertheless directly isogonal for appropriate pairs of rays.

Proof Let A be the point where f 01 ( z) is represented on the Kasner circle

off at z (Fig. 6.20a orb), and let 7/J = Argf 01 (z). Also, let B be the

second intersection of the secant OA with the circle. Call fJ 2 the value of

8 as the point f 0 (z) goes clockwise through B the first time around the

circle. If J > 0 (Fig. 6.20a) we shall take 82 as the direction associated

with 81. If J < 0 (Fig. 6.20b) we shall take 82 + 7r (second time around

through B) as the direction associated with 81.

For J > 0 we have

so that

and

Hence

19~ - 8~ = 82 - 81

In the case J < 0 we have

7/J = Arg !Bi( z) and

(a) (b)

Fig. 6.20

J<O