1550251515-Classical_Complex_Analysis__Gonzalez_

344 Chapter^6

The angle 7/J = Arg f8( z) is called the distortion angle at z.

Theorem 6.13 If a function has differentiable components at a point,

and if the directional derivatives there assume equal values for any two
incongruent directions (mod rr), the function is monogenic at that point.

Proof Suppose that at a point z we have

!8 1 (z) = f8 2 (z)

for any two directions 61 and 62 such that 62 i= 61 + krr whatever be the
integer k. This implies that

fz + fze-2i01 = fz + fze-2i02

and it follows that fz = 0. Hence f is monogenic at z.
In the following theorem the directional derivative of a complex function
f = u + iv is shown to be related to the real directional derivatives of its
components u and v.

Theorem 6.14 If f has differentiable components at a point z, then

(6.10-8)

where

Dou= Ux cos 6 +Uy sin6

Dov= Vx cos6 +Vy sin6 (6.10-9)

Proof In fact, equation (6.10-1) can also be written in the form

fe(z) = e-iO[( Ux cos 6 +Uy sin 6) + i(vx cos 6 +Vy sin 6)]

= e-iO(Dou + iDov)

The expression in parentheses may be thought as representing the rate of
change off with respect to distance in the direction specified by 6. To show
this, let ~z = jLlzjeio and keep 6 fixed while letting jLlzl -+ 0. We get

lim ( l~wl ) = lim ( ~w) ei

0

IAzl-+O t...:l.Z = Dou+ iDov

0 Az-+O t..J.Z 0

Theorem 6.15 If a function f has a vanishing directional derivative along

a regular arc 'Y, the derivative being taken at each point in the direction of
the arc, then the function is a constant along 'Y·

Proof If J 0 (z) = 0 for all z E 1*, from (6.10-1) we obtain

Uxx'(t) + uyy'(t) = 0 and Vxx'(t) + vyy'(t) = 0