1550251515-Classical_Complex_Analysis__Gonzalez_

Differentiation 379

so that

and

Hence

Thus in either case the mapping is directly isogonal.

We note that in the first case ( J > 0) we may use as well the directions

81 + 7r and 82 +'Tr, and in the second (J < 0) the directions 81 + 7r and (^82)
(i.e., the corresponding vertical angles).
To see what is the analytical relationship between 81 and 82 , let h be
the distance from the center of the circle to the secant OA (Fig. 6.20) and

let w = Arg fz. If h = 0 we have, trivially, 82 = 81 ±^1 / 2 7r. Suppose that

h f:. 0. From the geometry of the figure we obtain

Hence

cos(82 - 81 ) = :~:: sin(w - ~)

Corollary 6.3 Let f E 1J(N 0 (z)) and suppose that fz f:. 0, fz f:. 0, and
J f:. 0. Then the principal directions is the only pair of directions at z
which is mapped by f into an orthogonal pair at w = f(z).

Proof This is just a special case of Theorem 6.7, namely, when the secant

0 A passes through the center of the Kasner circle. It suffices to choose

along the principal directions the rays with arguments

The points f 01 ( z) and f 02 ( z) are the endpoints of the diameter of the circle

that contains the origin. If J > O, those are the only diametrically opposite

points for which~ = Argf~(z) has the same value. If J < O, they are
the only diametrically opposite points for which ~ = Arg JQ( z) has values
differing by ±7r.

6.19 SOME TRANSFORMATIONS INDUCED BY TiiE

DIRECTIONAL DERIVATIVE

As before, we assume that f E 1J(A) and let ( = e + iry = J 0 (z), z E A.
Clearly, we have the following induced mappings from A into the (-plane: