1550251515-Classical_Complex_Analysis__Gonzalez_

Differentiation 363

6.15 CONFORMAL MAPPINGS

In this section we discuss some elementary properties of the mappings de-

fined at a point by either analytic or conjugate analytic functions. In the

next three sections we consider the more general case of mappings defined

by functions of class '.D(A). A more detailed study of these mappings is

made in Selected Topics, Chapter 2.

Let w = f(z) E '.D(A), A open, and suppose that f is nonconstant in A.

In Section 6.10 we have seen that for each z E A we have

dw = f~(z) dz (6.15-1)

from which we derive

ldwl = IJ~(z)lldzl (6.15-2)

and

arg dw = Arg fMz) + Arg dz (6.15-3)

the preceding equation having a meaning for any direction (} such that

fMz) # 0 if f is polygenic at z. If f is monogenic at z, we must have

f'(z) # 0. We recall that the equality f~(z) = 0 can occur only if J1(z) = 0.

Let dz = ldzleill and dw = ldwlei^11 ', where B = Arg dz and B' is the

particular value of argdw defined by the right-hand side of (6.15-3). Also,
let f~(z) = pei.P so that p = lf 0 (z)I and 'I/;= Argf~(z) whenever f 0 (z) # 0.
Then equation (6.15-2) becomes

ldwl = pldzl

or

d<1 = pds

where ds = [dz[ and d<1 = ldw[, and (6.15-3) becomes

B' ='I/;+ B

(6.15-4)

(6.15-5)
As already noted in Section 6.10, the number pis called the magnification
ratio at the point z in the direction B, and 'I/;, which is called the distortion

angle, measures the amount of rotation experienced by dz under f. Both p

and 'I/; are functions of z and B. In what follows we propose to investigate
the nature of the dependence of p and 'ljJ on B at a fixed point z. In this

section we take up the cases in which f is either monogenic or conjugate

monogenic at z, whereas in the following two sections we discuss the case

of f a polygenic function at z.

If f is monogenic at z, both p and 'I/; are independent of B since fz = 0

and f 0 (z) reduces to the ordinary derivative f'(z), so that both p and 'I/;