388 Chapter^6
Theorem 6.38 If f is analytic in a region R and K is a compact subset
of R, then the increment ratio .6.f / .6.z tends uniformly to f^1 ( zo) for all
z 0 E K; i.e., for every given e > 0 there is a Ii > 0 such that
I
f(z) - f(zo) _ J'(zo)I < e
Z - Zofor all zo E Kand lz -zol < Ii.
Proof By Theorem 6.36 there exists a Ii > 0 such that
If((~={(,) -J'(zo)I < ~e
provided that I ( -zo I < 2/i and I>• - z 0 I < 2/i, and also such that
.IJ1(()-J'(zo)I <^1 / 2 e
whenever I( -z 0 I < 2/i (Theorem 6.37). Hence
(6.22-6)If((~={(,) -J'(()I ~If((~={(,) -f'(zo)I +If'(() -f'(zo)I < e
provided that I( -zol < Ii and I( -,\I < Ii, since the last two inequalities
imply that I,\ -zol ~ I( -,\I+ I( -z 01 < 2/i. This shows that there is an
open set containing z 0 and a Ii > 0 such that inequalities as (6.22-6) hold
on the open set. But K is compact, so it can be covered by a finite number
of such open sets. If li 1 , ... , lin are the corresponding i5's, the conclusion of
the theorem follows with o = min( '5 1 ... , lin)·
6.23 ON THE ANALYTICITY CONDITIONS
As we have seen (Theorem 6.5), a function w = f(z) = u(x,y) + iv(x,y)
defined in an open set D is analytic in D iff the functions u( x, y) and v( x, y)
are differentiable in D and satisfy the Cauchy-Riemann equations Ux = vy,
Uy = -Vx everywhere in D. However, the sufficiency of these conditions
in an open set are rather strong, and a number of authors, notably Pom-
peiu, Mantel, Looman, Menchoff, and Trokhimchuk, have sought to find
somewhat weaker conditions. In what follows we present a summary of
the results of those researchers referring the reader to the corresponding
papers, or to the Trokhimchuk monograph [121], for the detailed proofs.
D. Pompeiu [95] in 1905 was the first to prove that it suffices to assume
the continuity off and the existence of the derivative f' almost everywhere
in D (i.e., except in a subset of zero measure), provided that the limit
-1· Im 1.6.w - I
~z->O .6.z(6.23-1)