1550251515-Classical_Complex_Analysis__Gonzalez_

Differentiation 365

the mapping when f is analytic in R (with f' -:/= 0), a small figure in R

and the corresponding image in f(R) are approximately similar. Thus the

form of small figures are nearly preserved under f, and the mapping is

said to be conformal (more precisely, directly conformal or sensepreserving

conformal'].

If f is conjugate monogenic at z, then fz = 0, so that f~(z) = fze-^2 i1J

and the magnification ratio

p = IJ~(z)I = lf::I = l]'(z)I

is still constant at a fixed point z. If fz -:/= 0 and a = Arg Jz, we have

.,P = Argf~(z) =a - 20 + 2k7r

for an appropriate value of the integer k. Hence substituting in (6.15-5),
we have
O' = a - 0 + 2k7r
For two different directions (}i and 02 we obtain
and O~ =a - 82 + 2k2?T

so that

o~ -o~ = -( 02 - 01) (mod2?T)

Hence in this case the magnitude of the angle between two directions is

preserved but its orientation is reversed. Mappings having this property

are called indirectly isogonal. If f is conjugate analytic in some region R,

and fz -:/= 0 on R, the mapping defined by f is said to be indirectly isogonal

in R. Since p( z) = I]' ( z) I is again continuous in R, the form of small figures

in R is approximately preserved under f. Because of this the mapping is

called indirectly conformal on R (or sense-reversing conformal'].

The simplest indirectly conformal mapping is provided by the conjuga-

tion function g(z) = z. Geometrically, this is a reflection on the real axis.

From this and the identity f = J it follows also that the mapping defined

by any conjugate analytic function f with fz -:/= 0 is indirectly conformal.

In fact, the mapping z--> f(z) is directly conformal(] being analytic), and
the mapping g: ](z) --> ](z) = f(z) is indirectly conformal, so that the
composition go ](z) = f(z) is then indirectly conformal (Fig. 6.10).
In the two cases discussed above the magnification ratio p is constant at a
fixed point z. We may ask whether those are the only two cases for which
p is constant. An affirmative analytical proof was given in Section 6.10

(Theorem 6.17). It is easier to see, by considering the corresponding Kasner

circle, that there are no other cases. In fact, since ( = f~( z) is on the Kasner

circle, its modulus (i.e., its distance from the origin) is constant i:ff (1) the