3Residue calculus

IB Complex Analysis



3.2 Homotopy of closed curves
The last ingredient we need before we can get to the residue theorem is the
idea of homotopy. Recall we had this weird, ugly definition of elementary
deformation of curves given
φ, ψ
: [
a, b
]
U
, which are closed, we say
ψ
is an
elementary deformation or convex deformation of
φ
if there exists a decomposition
a
=
x
0
< x
1
< ··· < x
n
=
b
and convex open sets
C
1
, ··· , C
n
U
such that for
x
i1
t x
i
, we have φ(t) and ψ(t) in C
i
.
It was a rather unnatural definition, since we have to make reference to
this arbitrarily constructed dissection of [
a, b
] and convex sets
C
i
. Moreover,
this definition fails to be transitive (e.g. on
R \{
0
}
, rotating a circle about the
center by, say,
π
10
is elementary, but rotating by
π
is not). Yet, this definition
was cooked up just so that it immediately follows that elementary deformations
preserve integrals of holomorphic functions around the loop.
The idea now is to define a more general and natural notion of deforming a
curve, known as “homotopy”. We will then show that each homotopy can be
given by a sequence of elementary deformations. So homotopies also preserve
integrals of holomorphic functions.
Definition
(Homotopy of closed curves)
.
Let
U C
be a domain, and let
φ
: [
a, b
]
U
and
ψ
: [
a, b
]
U
be piecewise
C
1
-smooth closed paths. A
homotopy from φ : ψ is a continuous map F : [0, 1] ×[a, b] U such that
F (0, t) = φ(t), F (1, t) = ψ(t),
and moreover, for all
s
[0
, t
], the map
t 7→ F
(
s, t
) viewed as a map [
a, b
]
U
is closed and piecewise C
1
-smooth.
We can imagine this as a process of “continuously deforming” the path
φ
to
ψ, with a path F (s, ·) at each point in time s [0, 1].
Proposition.
Let
φ, ψ
: [
a, b
]
U
be homotopic (piecewise
C
1
) closed paths in
a domain
U
. Then there exists some
φ
=
φ
0
, φ
1
, ··· , φ
N
=
ψ
such that each
φ
j
is piecewise
C
1
closed and
φ
i+1
is obtained from
φ
i
by elementary deformation.
Proof.
This is an exercise in uniform continuity. We let
F
: [0
,
1]
×
[
a, b
]
U
be a homotopy from
φ
to
ψ
. Since
image
(
F
) is compact and
U
is open, there
is some
ε >
0 such that
B
(
F
(
s, t
)
, ε
)
U
for all (
s, t
)
[0
,
1]
×
[
a, b
] (for each
s, t
, pick the maximum
ε
s,t
>
0 such that
B
(
F
(
s, t
)
, ε
s,t
)
U
. Then
ε
s,t
varies
continuously with
s, t
, hence attains its minimum on the compact set [0
,
1]
×
[
a, b
].
Then picking ε to be the minimum works).
Since
F
is uniformly continuous, there is some
δ
such that
k
(
s, t
)
(
s
0
, t
0
)
k < δ
implies |F (s, t) F (s
0
, t
0
)| < ε.
Now we pick n N such that
1+(ba)
n
< δ, and let
x
j
= a + (b a)
j
n
φ
i
(t) = F
i
n
, t
C
ij
= B
F
i
n
, x
j
, ε
Then
C
ij
is clearly convex. These definitions are cooked up precisely so that if
s
i1
n
,
i
n
and t [x
j1
, x
j
], then F (s, t) C
ij
. So the result follows.
Corollary.
Let
U
be a domain,
f
:
U C
be holomorphic, and
γ
1
, γ
2
be
homotopic piecewise C
1
-smooth closed curves in U. Then
Z
γ
1
f(z) dz =
Z
γ
2
f(z) dz.
This means the integral around any path depends only on the homotopy
class of the path, and not the actual path itself.
We can now use this to “upgrade” our Cauchy’s theorem to allow arbitrary
simply connected domains. The theorem will become immediate if we adopt the
following alternative definition of a simply connected domain:
Definition
(Simply connected domain)
.
A domain
U
is simply connected if
every C
1
smooth closed path is homotopic to a constant path.
This is in fact equivalent to our earlier definition that every continuous
map
S
1
U
can be extended to a continuous map
D
2
U
. This is almost
immediately obvious, except that our old definition only required the map to be
continuous, while the new definition only works with piecewise
C
1
paths. We
will need something that allows us to approximate any continuous curve with a
piecewise
C
1
-smooth one, but we shall not do that here. Instead, we will just
forget about the old definition and stick to the new one.
Rewriting history, we get the following corollary:
Corollary
(Cauchy’s theorem for simply connected domains)
.
Let
U
be a simply
connected domain, and let
f
:
U C
be holomorphic. If
γ
is any piecewise
C
1
-smooth closed curve in U, then
Z
γ
f(z) dz = 0.
We will sometimes refer to this theorem as “simply-connected Cauchy”, but
we are not in any way suggesting that Cauchy himself is simply connected.
Proof.
By definition of simply-connected,
γ
is homotopic to the constant path,
and it is easy to see the integral along a constant path is zero.