1550251515-Classical_Complex_Analysis__Gonzalez_

Chapter6

are constants at a fixed point whenever f is monogenic at that point [and

J'(z) f= 0 in referring to 1/J].
Let dz 1 and dz 2 be two arbitrary increments from the point z, with

(Ji = Argdz1, 02 = Argdz2, and let dw1 = J'(z) dz1, dw2 = f'(z)dz2. If

Oi and !9~ are the arguments of dw 1 and dw 2 , respectively, as determined
by (6.15-3), we have

and

Hence

o; -Oi = 02 - 01 ·

which shows that the angle between the directions of dz 1 and dz 2 is pre-
served in magnitude and orientation under the mapping defined by a
function f that is monogenic at z, provided that f'(z) f= 0. This implies
that if two smooth arcs '/'l and '}' 2 intersect at z and the corresponding

semitangents at z form an angle 02 - 01 , their images under f, namely,

rl = !(11) and r2 = f(1'2), are smooth arcs intersecting at w = f(z)
and such that the corresponding semitangents at that point form an angle
O~ - Oi which equals 02 - (Ji. (Fig. 6.9). Briefly, whenever f'(z) f= 0 the
angle between two smooth arcs intersecting at z is preserved, in size as well
as orientation, under the mapping defined by a monogenic function J(z).

Mappings having this property are called directly isogonal. If f is ana-

lytic in a region R, and f' ( z) f= 0 on R, then the mapping defined by f
is directly isogonal in R.

As we have pointed out, the magnification ratio p(z) = IJ'(z)I is con-

stant at a fixed point z. If f is analytic in a region R, then p(z) is a

continuqus function of z, since f' is then continuous in R, as we prove
later (see Sections 6.22 and 7.21). Because of this, and the isogonality of

y v

(^0) x 0 u
Fig. 6.9