Cauchys theorem

24.9 CAUCHY’S THEOREM
namely Cauchy’s theorem, which is the cornerstone of the integral calculus of
complex variables.
Before discussing Cauchy’s theorem, however, we note an important result
concerning complex integrals that will be of some use later. Let us consider the
integral of a function f(z) along some path C. If M is an upper bound on the
value of |f(z)| on the path, i.e. |f(z)| ≤ M on C, and L is the length of the path C,
then
f(z) dz ≤ |f(z)||dz| ≤ M
dl = ML.
(24.39)
C
c
C
It is straightforward to verify that this result does indeed hold for the complex
integrals considered earlier in this section.
24.9 Cauchy’s theorem
Cauchy’s theorem states that if f(z) is an analytic function, and f (z) is continuous
at each point within and on a closed contour C, then
0
f(z) dz = 0.
(24.40)
C
In this
/ statement and from now on we denote an integral around a closed contour
by C .
To prove this theorem we will need the two-dimensional form of the divergence
theorem, known as Green’s theorem in a plane (see section 11.3). This says that
if p and q are two functions with continuous first derivatives within and on a
closed contour C (bounding a domain R) in the xy-plane, then
0
∂p ∂q
+
(24.41)
dxdy = (p dy − q dx).
∂x ∂y
R
C
With f(z) = u + iv and dz = dx + i dy, this can be applied to
0
0
0
f(z) dz = (u dx − v dy) + i (v dx + u dy)
I=
C
C
C
to give
I=
R
∂(−u) ∂(−v)
∂(−v) ∂u
+
dx dy + i
+
dx dy.
∂y
∂x
∂y
∂x
R
(24.42)
Now, recalling that f(z) is analytic and therefore that the Cauchy–Riemann
relations (24.5) apply, we see that each integrand is identically zero and thus I is
also zero; this proves Cauchy’s theorem.
In fact, the conditions of the above proof are more stringent than they need
be. The continuity of f (z) is not necessary for the proof of Cauchy’s theorem,
849
COMPLEX VARIABLES
y
B
C1
R
x
A
C2
Figure 24.10 Two paths C1 and C2 enclosing a region R.
analyticity of f(z) within and on C being sufficient. However, the proof then
becomes more complicated and is too long to be given here.§
The connection between Cauchy’s theorem and the zero value of the integral
of z −1 around the composite path C4 discussed towards the end of the previous
section is apparent: the function z −1 is analytic in the two regions of the z-plane
enclosed by contours (C2 and C3a ) and (C2 and C3b ).
Suppose two points A and B in the complex plane are joined by two different paths C1
and C2 . Show that if f(z) is an analytic function on each path and in the region enclosed
by the two paths, then the integral of f(z) is the same along C1 and C2 .
The situation is shown in figure 24.10. Since f(z) is analytic in R, it follows from Cauchy’s
theorem that we have
0
f(z) dz −
f(z) dz =
f(z) dz = 0,
C1
C1 −C2
C2
since C1 − C2 forms a closed contour enclosing R. Thus we immediately obtain
f(z) dz =
f(z) dz,
C1
C2
and so the values of the integrals along C1 and C2 are equal. An important application of Cauchy’s theorem is in proving that, in some
cases, it is possible to deform a closed contour C into another contour γ in such
a way that the integrals of a function f(z) around each of the contours have the
same value.
§
The reader may refer to almost any book that is devoted to complex variables and the theory of
functions.
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