1550251515-Classical_Complex_Analysis__Gonzalez_

342 Chapter 6

(96], equals fz except for the constant factor 2i, assuming f to be of class

C(l)(A). For f continuous on A, C a simple closed contour surrounding

z E A, and a = area of the region R bounded by C, the Pompeiu derivative

is defined by

lim ]:_ [ f(z)dz

.,. ..... o CT Jc

whenever the limit exists, the limit being taken as C ultimately lies within

an arbitrarily small circle about z. The integral above may be defined in

terms of real line integrals as follows:

l f(z) dz= l(udx - vdy) ti l(v dx t-udy)

On applying Green's theorem to eac~ of the line integrals on the right,

we have

[ f(z) dz= f k(-vx - uy)dx 41;! +if l(Y.?' -vy) dx dy

= 2i J k %[('4f :-: vy) + i(vx + u11.)] dx dy

= 2i f kt~~;fJ4.'!j

The law of the mean for douple integrals yit:)lps

l f(z) dz==-?if~(() J k dx qy = 2ifz(()a

for a suitable value of ( in ~' As a --+ 0, ( --+ z, and we obtain

lim l [ f(z) dz= 2ifz(z)

u-+o a Jc

Clearly, when fz = 0 the directional derivative is independent of (}, and

f~(z) = fz(z) = f'(z), i.e., th<=; function is monogenic at z.

If fz =f:. O, then the direction.al derivative of f at z depends on (}, and

f is said to be polygenic at th~ point (Kasner). The value lfzl has been
called the deviation from analyticity by Szu-Hoa Min (115].
In the case of a polygenic function the directional derivative at z depends
on the direction of the arc at z, i.e., on the orientation of the semitangent
line at z, in the sense of the increasing arc, and not on any other property
of the arc (supposedly regular). Note tl;iat the opposite sernitangent at z
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