3Residue calculus
IB Complex Analysis
3.1 Winding numbers
Recall that the type of the singularity of a point depends on the coefficients in
the Laurent series, and these coefficients play an important role in determining
the behaviour of the functions. Among all the infinitely many coefficients, it
turns out the coefficient of
z
−1
is the most important one, as we will soon see.
We call this the residue of f.
Definition
(Residue)
.
Let
f
:
B
(
a, r
)
\ {a} → C
be holomorphic, with Laurent
series
f(z) =
∞
X
n=−∞
c
n
(z −a)
n
.
Then the residue of f at a is
Res(f, a) = Res
f
(a) = c
−1
.
Note that if ρ < r, then by definition of the Laurent coefficients, we know
Z
∂B(a,ρ)
f(z) dz = 2πic
−1
.
So we can alternatively write the residue as
Res
f
(a) =
1
2πi
Z
∂B(a,ρ)
f(z) dz.
This gives us a formulation of the residue without reference to the Laurent series.
Deforming paths if necessary, it is not too far-fetching to imagine that for
any simple curve γ around the singularity a, we have
Z
γ
f(z) dz = 2πi Res(f, a).
Moreover, if the path actually encircles two singularities
a
and
b
, then deforming
the path, we would expect to have
Z
γ
f(z) dz = 2πi(Res(f, a) + Res(f, b)),
and this generalizes to multiple singularities in the obvious way.
If this were true, then it would be very helpful, since this turns integration
into addition, which is (hopefully) much easier!
Indeed, we will soon prove that this result holds. However, we first get rid of
the technical restriction that we only work with simple (i.e. non-self intersecting)
curves. This is completely not needed. We are actually not really worried in
the curve intersecting itself. The reason why we’ve always talked about simple
closed curves is that we want to avoid the curve going around the same point
many times.
There is a simple workaround to this problem — we consider arbitrary curves,
and then count how many times we are looping around the point. If we are
looping around it twice, then we count its contribution twice!
Indeed, suppose we have the following curve around a singularity:
a
We see that the curve loops around
a
twice. Also, by the additivity of the
integral, we can break this curve into two closed contours. So we have
1
2πi
Z
γ
f(z) dz = 2 Res
f
(a).
So what we want to do now is to define properly what it means for a curve to
loop around a point n times. This will be called the winding number.
There are many ways we can define the winding number. The definition we
will pick is based on the following observation — suppose, for convenience, that
the point in question is the origin. As we move along a simple closed curve around
0, our argument will change. If we keep track of our argument continuously,
then we will find that when we return to starting point, the argument would
have increased by 2
π
. If we have a curve that winds around the point twice,
then our argument will increase by 4π.
What we do is exactly the above — given a path, find a continuous function
that gives the “argument” of the path, and then define the winding number to
be the difference between the argument at the start and end points, divided by
2π.
For this to make sense, there are two important things to prove. First, we
need to show that there is indeed a continuous “argument” function of the curve,
in a sense made precise in the lemma below. Then we need to show the winding
number is well-defined, but that is easier.
Lemma.
Let
γ
: [
a, b
]
→ C
be a continuous closed curve, and pick a point
w ∈ C \ image
(
γ
). Then there are continuous functions
r
: [
a, b
]
→ R >
0 and
θ : [a, b] → R such that
γ(t) = w + r(t)e
iθ(t)
.
Of course, at each point
t
, we can find
r
and
θ
such that the above holds.
The key point of the lemma is that we can do so continuously.
Proof.
Clearly
r
(
t
) =
|γ
(
t
)
− w|
exists and is continuous, since it is the composi-
tion of continuous functions. Note that this is never zero since
γ
(
t
) is never
w
.
The actual content is in defining θ.
To define
θ
(
t
), we for simplicity assume
w
= 0. Furthermore, by considering
instead the function
γ(t)
r(t)
, which is continuous and well-defined since
r
is never
zero, we can assume |γ(t)| = 1 for all t.
Recall that the principal branch of
log
, and hence of the argument
Im
(
log
),
takes values in (−π, π) and is defined on C \ R
≤0
.
If
γ
(
t
) always lied in, say, the right-hand half plane, we would have no problem
defining θ consistently, since we can just let
θ(t) = arg(γ(t))
for
arg
the principal branch. There is nothing special about the right-hand half
plane. Similarly, if γ lies in the region as shaded below:
α
i.e. we have
γ(t) ∈
n
z : Re
z
e
iα
> 0
o
for a fixed α, we can define
θ(t) = α + arg
γ(t)
e
iα
.
Since γ : [a, b] → C is continuous, it is uniformly continuous, and we can find a
subdivision
a = a
0
< a
1
< ··· < a
m
= b,
such that if
s, t ∈
[
a
i−1
, a
i
], then
|γ
(
s
)
− γ
(
t
)
| <
√
2
, and hence
γ
(
s
) and
γ
(
t
)
belong to such a half-plane.
So we define θ
j
: [a
j−1
, a
j
] → R such that
γ(t) = e
iθ
j
(t)
for t ∈ [a
j−1
, a
j
], and 1 ≤ j ≤ n − 1.
On each region [
a
j−1
, a
j
], this gives a continuous argument function. We
cannot immediately extend this to the whole of [
a, b
], since it is entirely possible
that
θ
j
(
a
j
) =
θ
j+1
(
a
j
). However, we do know that
θ
j
(
a
j
) are both values of the
argument of
γ
(
a
j
). So they must differ by an integer multiple of 2
π
, say 2
nπ
.
Then we can just replace
θ
j+1
by
θ
j+1
−
2
nπ
, which is an equally valid argument
function, and then the two functions will agree at a
j
.
Hence, for
j >
1, we can successively re-define
θ
j
such that the resulting map
θ is continuous. Then we are done.
We can thus use this to define the winding number.
Definition
(Winding number)
.
Given a continuous path
γ
: [
a, b
]
→ C
such
that γ(a) = γ(b) and w 6∈ image(γ), the winding number of γ about w is
θ(b) −θ(a)
2π
,
where
θ
: [
a, b
]
→ R
is a continuous function as above. This is denoted by
I
(
γ, w
)
or n
γ
(W ).
I and n stand for index and number respectively.
Note that we always have
I
(
γ, w
)
∈ Z
, since
θ
(
b
) and
θ
(
a
) are arguments
of the same number. More importantly,
I
(
γ, w
) is well-defined — suppose
γ
(
t
) =
r
(
t
)
e
iθ
1
(t)
=
r
(
t
)
e
iθ
2
(t)
for continuous functions
θ
1
, θ
2
: [
a, b
]
→ R
. Then
θ
1
− θ
2
: [
a, b
]
→ R
is continuous, but takes values in the discrete set 2
πZ
. So it
must in fact be constant, and thus θ
1
(b) − θ
1
(a) = θ
2
(b) − θ
2
(a).
So far, what we’ve done is something that is true for arbitrary continuous
closed curve. However, if we focus on piecewise
C
1
-smooth closed path, then we
get an alternative expression:
Lemma.
Suppose
γ
: [
a, b
]
→ C
is a piecewise
C
1
-smooth closed path, and
w 6∈ image(γ). Then
I(γ, w) =
1
2πi
Z
γ
1
z − w
dz.
Proof. Let γ(t) − w = r(t)e
iθ(t)
, with now r and θ piecewise C
1
-smooth. Then
Z
γ
1
z − w
dz =
Z
b
a
γ
0
(t)
γ(t) − w
dt
=
Z
b
a
r
0
(t)
r(t)
+ iθ
0
(t)
dt
= [ln r(t) + iθ(t)]
b
a
= i(θ(b) −θ(a))
= 2πiI(γ, w).
So done.
In some books, this integral expression is taken as the definition of the
winding number. While this is elegant in complex analysis, it is not clear a priori
that this is an integer, and only works for piecewise
C
1
-smooth closed curves,
not arbitrary continuous closed curves.
On the other hand, what is evident from this expression is that
I
(
γ, w
) is
continuous as a function of
w ∈ C \ image
(
γ
), since it is even holomorphic as a
function of
w
. Since
I
(
γ
;
w
) is integer valued,
I
(
γ
) must be locally constant on
path components of C \ image(γ).
We can quickly verify that this is a sensible definition, in that the winding
number around a point “outside” the curve is zero. More precisely, since
image
(
γ
)
is compact, all points of sufficiently large modulus in
C
belong to one component
of
C \image
(
γ
). This is indeed the only path component of
C \image
(
γ
) that is
unbounded.
To find the winding number about a point in this unbounded component, note
that
I
(
γ
;
w
) is consistent on this component, and so we can consider arbitrarily
larger w. By the integral formula,
|I(γ, w)| ≤
1
2π
length(γ) max
z∈γ
1
|w − z|
→ 0
as
w → ∞
. So it does vanish outside the curve. Of course, inside the other path
components, we can still have some interesting values of the winding number.