(^198) Functions of a Complex Variable
the best proof, because the definition of an analytic function requires only
the existence of the derivative.
In 1900, the French mathematician EDOUARD GOURSAT (1858-1936)
produced a proof that does not rely on the continuity of /'. He published
the result in 1900 in the very first issue of the Transactions of the American
Mathematical Society (Volume 1, No. 1, pages 14-16; the paper is in
French, though). Goursat's proof is too technical to discuss here and is
more suitable for a graduate-level course in complex analysis.
The converse statement (path independence implies analyticity) was
proved by the Italian mathematician GIACINTO MORERA (1856-1909). We
almost have the proof of this result too: with Theorem 4.2.2 at our disposal,
all we need is the infinite differentiability of analytic functions, which we
will establish later in this section.
Remark 4.2 Assume that the function f is continuous in the closure
of G (that is, f is defined on the boundary of G and, for every ZQ on the
boundary of G, f{z) —> f(zo) as z —> ZQ, Z 6 G.) If f is analytic in G
and G is simply connected and bounded with a piece-wise smooth boundary
Co, then §c f{z)dz = 0. Indeed, continuity of f implies that, for every
e > 0, there exists a simple, closed, piece-wise smooth path Ce lying inside
of G such that | §c f(z)dz — §c f{z)dz\ < s. By the Integral Theorem of
Cauchy, §c f{z)dz — 0.
The Integral Theorem of Cauchy says nothing about domains that are
not simply connected; Exercise 4.2.13, in which G is the disk with the center
removed, shows that anything can happen in those domains. If the holes
in G are sufficiently nice, then we can be more specific.
Theorem 4.2.4 Assume that the boundary of G consists ofn+1 closed,
simple, piece-wise smooth curves CQ, ... ,Cn so that C\,... ,Cn do not have
points in common and are all inside the domain enclosed by CQ. Assume
that f is a function analytic in G and is continuous in the closure of G.
Then
I f(z)dz + J2(£ f(z)dz = 0, (4.2.10)
J Co k=1 JCk
where the curve CQ is oriented counterclockwise, and all other curves, clock-
wise, so that, if you walk in the direction of the orientation, the domain G
is always on the left.
coco
(coco)
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