3Residue calculus
IB Complex Analysis
3.4 Overview
We’ve done most of the theory we need. In the remaining of the time, we are
going to use these tools to do something useful. In particular, we will use the
residue theorem heavily to compute integrals.
But before that, we shall stop and look at what we have done so far.
Our first real interesting result was Cauchy’s theorem for a triangle, which
had a rather weird hypothesis — if
f
:
U → C
is holomorphic and ∆
⊆ U
is at
triangle, then
Z
∂∆
f(z) dz = 0.
To prove this, we dissected our triangle into smaller and smaller triangles, and
then the result followed how the numbers and bounds magically fit in together.
To accompany this, we had another theorem that used triangles. Suppose
U
is a star domain and f : U → C is continuous. Then if
Z
∂∆
f(z) dz = 0
for all triangles, then there is a holomorphic
F
with
F
0
(
z
) =
f
(
z
). Here we
defined F by
F (z) =
Z
z
z
0
f(z) dz,
where
z
0
is the “center” of the star, and we integrate along straight lines. The
triangle condition ensures this is well-defined.
These are the parts where we used some geometric insight — in the first
case we thought of subdividing, and in the second we decided to integrate along
paths.
These two awkward theorems about triangles fit in perfectly into the convex
Cauchy theorem, via the fundamental theorem of calculus. This tells us that if
f : U → C is holomorphic and U is convex, then
Z
γ
f(z) dz = 0
for all closed γ ⊆ U .
We then noticed this allows us to deform paths nicely and still preserve the
integral. We called these nice deformations elementary deformations, and then
used it to obtain the Cauchy integral formula, namely
f(w) =
1
2πi
Z
∂B(a,ρ)
f(z)
z −w
dz
for f : B(a, r) → C, ρ < r and w ∈ B(a, ρ).
This formula led us to some classical theorems like the Liouville theorem and
the maximum principle. We also used the power series trick to prove Taylor’s
theorem, saying any holomorphic function is locally equal to some power series,
which we call the Taylor series. In particular, this shows that holomorphic
functions are infinitely differentiable, since all power series are.
We then notice that for
U
a convex domain, if
f
:
U → C
is continuous and
Z
γ
f(z) dz = 0
for all curves
γ
, then
f
has an antiderivative. Since
f
is the derivative of its
antiderivative (by definition), it is then (infinitely) differentiable. So a function
is holomorphic on a simply connected domain if and only if the integral along
any closed curve vanishes. Since the latter property is easily shown to be
conserved by uniform limits, we know the uniform limit of holomorphic functions
is holomorphic.
Then we figured out that we can use the same power series expansion trick
to deal with functions with singularities. It’s just that we had to include
negative powers of
z
. Adding in the ideas of winding numbers and homotopies,
we got the residue theorem. We showed that if
U
is simply connected and
f : U \ {z
1
, ··· , z
k
} → C is holomorphic, then
1
2πi
Z
γ
f(z) dz =
X
Res(f, z
i
)I(γ, z
i
).
This will further lead us to Rouch´e’s theorem and the argument principle, to be
done later.
Throughout the course, there weren’t too many ideas used. Everything was
built upon the two “geometric” theorems of Cauchy’s theorem for triangles and
the antiderivative theorem. Afterwards, we repeatedly used the idea of deforming
and cutting paths, as well as the power series expansion of
1
z−w
, and that’s it.