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20.2 Cauchy’s Theorem 701
Exterior
y
Interior
x
FIGURE 20.5 Interior and exterior
of a closed curve.
We will refer to a simple, piecewise smooth curve as a path. A path in a set S is a path
whose graph lies in S.
Aset S of complex numbers is connected if every two points of S are endpoints of a path
in S. This means that we can get from any point of S to any other point by moving along
some path without leaving S. An open, connected set is called a domain. For example, any
open disk is a domain, and the right quarter plane x > 0, y > 0 is a domain.
We encountered domains in connection with potential functions in Chapter 12.
Aset S of complex numbers is simply connected if every closed path in S encloses only
points of S.
This concept was also discussed in Chapter 12. Every open disk is simply connected. However,
let S be an open disk with the center removed (a punctured disk). Then a closed path about the
center in the disk encloses a point not in the set, so this set is not simply connected (although it
is open and connected, hence is a domain).
We can now state the main result.
THEOREM 20.1 Cauchy
Let f be differentiable on a simply connected domain G. Then
f (z)dz = 0
γ
for every closed path γ in G.
Cauchy’s theorem means that f (z)dz = 0if f is differentiable on the path γ and all
γ
points enclosed by γ . Unless otherwise stated, we always understand closed curves to be oriented
counterclockwise, which we take to be the positive sense.
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