Complex Analysis - Maths KU

D C
Figure 5.26 Simply connected
domain D
C 1
D
C 2
Figure 5.27 Multiply connected
domain D
Green’s theorem expresses a real
line integral as a double integral
☞
5.3 Cauchy-Goursat Theorem 257
D. Expressed yet another way, a simply connected domain has no “holes” in
it. The entire complex plane is an example of a simply connected domain;
the annulus defined by 1 < |z| < 2 is not simply connected. (Why?) A domain
that is not simply connected is called a multiply connected domain;
that is, a multiply connected domain has “holes” in it. Note in Figure 5.27
that if the curve C2 enclosing the “hole” were shrunk to a point, the curve
would have to leave D eventually. We call a domain with one “hole” doubly
connected, a domain with two “holes” triply connected, and so on. The
open disk defined by |z| < 2 is a simply connected domain; the open circular
annulus defined by 1 < |z| < 2 is a doubly connected domain.
Cauchy’s Theorem In 1825 the French mathematician Louis-Augustin
Cauchy proved one the most important theorems in complex analysis.
Cauchy’s Theorem
Suppose that a function f is analytic in a simply connected domain
D and that f ′ is continuous in D. Then for every simple closed (1)
contour C in D, �
f(z) dz =0.
C
Cauchy’s Proof of (1) The proof of this theorem is an immediate consequence
of Green’s theoremin the plane and the Cauchy-Riemann equations.
Recall fromcalculus that if C is a positively oriented, piecewise smooth, simple
closed curve forming the boundary of a region R within D, and if the
real-valued functions P (x, y) and Q(x, y) along with their first-order partial
derivatives are continuous on a domain that contains C and R, then
�
C
�� � �
∂Q ∂P
Pdx+ Qdy = − dA. (2)
R ∂x ∂y
Now in the statement (1) we have assumed that f ′ is continuous throughout
the domain D. As a consequence, the real and imaginary parts of f(z) =u+iv
and their first partial derivatives are continuous throughout D. By (9) of
Section 5.2 we write �
f(z) dz in terms of real line integrals and apply Green’s
C
theorem(2) to each line integral:
�
C
�
�
f(z) dz = u(x, y) dx − v(x, y) dy + i
C
��
=
�
− ∂v
� ��
∂u
− dA + i
∂x ∂y
R
C
R
v(x, y) dx + u(x, y) dy
� �
∂u ∂v
− dA. (3)
∂x ∂y
Because f is analytic in D, the real functions u and v satisfy the Cauchy-
Riemann equations, ∂u/∂x = ∂v/∂y and ∂u/∂y = −∂v/∂x, at every point in