2Contour integration and Cauchy's theorem
IB Complex Methods
2.2 Cauchy’s theorem
We now come to the highlight of the course — Cauchy’s theorem. Most of the
things we do will be based upon this single important result.
Theorem
(Cauchy’s theorem)
.
If
f
(
z
) is analytic in a simply-connected domain
D, then for every simple closed contour γ in D, we have
I
γ
f(z) dz = 0.
This is quite a powerful statement, and will allow us to do a lot! On the other
hand, this tells us functions that are analytic everywhere are not too interesting.
Instead, we will later look at functions like
1
z
that have singularities.
Proof.
(non-examinable) The proof of this remarkable theorem is simple (with a
catch), and follows from the Cauchy-Riemann equations and Green’s theorem.
Recall that Green’s theorem says
I
∂S
(P dx + Q dy) =
ZZ
S
∂Q
∂x
−
∂P
∂y
dx dy.
Let u, v be the real and imaginary parts of f . Then
I
γ
f(z) dz =
I
γ
(u + iv)(dx + i dy)
=
I
γ
(u dx − v dy) + i
I
γ
(v dx + u dy)
=
ZZ
S
−
∂v
∂x
−
∂u
∂y
dx dy + i
ZZ
S
∂u
∂x
−
∂v
∂y
dx dy
But both integrands vanish by the Cauchy-Riemann equations, since
f
is differ-
entiable throughout S. So the result follows.
Actually, this proof requires
u
and
v
to have continuous partial derivatives
in
S
, otherwise Green’s theorem does not apply. We shall see later that in fact
f
is differentiable infinitely many time, so actually
u
and
v
do have continuous
partial derivatives. However, our proof of that will utilize Cauchy’s theorem! So
we are trapped.
Thus a completely different proof (and a very elegant one!) is required if we
do not wish to make assumptions about
u
and
v
. However, we shall not worry
about this in this course since it is easy to verify that the functions we use do
have continuous partial derivatives. And we are not doing Complex Analysis.