This section presents the general version of the Cauchy integral formula valid for
arbitrary closed rectifiable curves. The key idea in this development is the notion of the
winding number. This is the number also called the index, defined in the following
theorem. This winding number, along with the earlier results, especially Liouville’s
theorem, yields an extremely general Cauchy integral formula.
Definition 50.7.14Let γ :
[a,b]
→ ℂ and suppose z
∕∈
γ^{∗}. The winding number, n
(γ,z)
is defined by
∫
n (γ,z) ≡ -1- -dw--.
2πi γ w − z
The main interest is in the case where γ is closed curve. However, the same notationwill be used for any such curve.
Theorem 50.7.15Let γ :
[a,b]
→ ℂ be continuous and have bounded variation withγ
(a)
= γ
(b)
. Also suppose that z
∕∈
γ^{∗}. Define
∫
n (γ,z) ≡ -1- -dw--. (50.7.23)
2πi γ w − z
(50.7.23)
Then n
(γ,⋅)
is continuous and integer valued. Furthermore, there exists a sequence,η_{k} :
[a,b]
→ ℂ such that η_{k}is C^{1}
([a,b])
,
1
||ηk − γ|| < k-,ηk(a) = ηk(b) = γ(a) = γ (b),
and n
(ηk,z)
= n
(γ,z)
for all k large enough. Also n
(γ,⋅)
is constant on everyconnected component ofℂ∖γ^{∗}and equals zero on the unbounded component ofℂ∖γ^{∗}.
Proof: First consider the assertion about continuity.
|∫ ( ) |
|| --1-- ---1-- ||
|n(γ,z)− n(γ,z1)| ≤ C | γ w − z − w − z1 dw|
^
≤ C (Length of γ)|z1 − z|
whenever z_{1} is close enough to z. This proves the continuity assertion. Note this did not
depend on γ being closed.
Next it is shown that for a closed curve the winding number equals an integer. To do
so, use Theorem 49.0.12 to obtain η_{k}, a function in C^{1}
= 2mπi for some integer m.
Therefore, 50.7.24 implies
∫ b η′(s)ds ∫ dw
2m πi = η-(s)-− z-= w-− z-.
a η
Therefore,
1--
2πi
∫_{ηk}
-dw--
w−z
is a sequence of integers converging to
1--
2πi
∫_{γ}
-dw-
w−z
≡ n
(γ,z)
and so n
(γ,z)
must also be an integer and n
(ηk,z)
= n
(γ,z)
for all k large
enough.
Since n
(γ,⋅)
is continuous and integer valued, it follows from Corollary 6.9.11 on
Page 494 that it must be constant on every connected component of ℂ∖γ^{∗}. It is
clear that n
(γ,z)
equals zero on the unbounded component because from the
formula,
( 1 )
zli→m∞|n(γ,z)| ≤ lzi→m∞ V (γ,[a,b]) ------
|z|− c
. This proves the corollary since the assertion about continuity
was already observed.
It is a good idea to consider a simple case to get an idea of what the winding number
is measuring. To do so, consider γ :
[a,b]
→ ℂ such that γ is continuous, closed and
bounded variation. Suppose also that γ is one to one on
(a,b)
. Such a curve is called a
simple closed curve. It can be shown that such a simple closed curve divides the
plane into exactly two components, an “inside” bounded component and an
“outside” unbounded component. This is called the Jordan Curve theorem. This is a
difficult theorem which requires some very hard topology such as homology
theory or degree theory. It won’t be used here beyond making reference to it. For
now, it suffices to simply assume that γ is such that this result holds. This
will usually be obvious anyway. Also suppose that it is possible to change the
parameter to be in
[0,2π]
, in such a way that γ
(t)
+ λ
( it )
z + re − γ (t)
− z≠0 for
all t ∈
[0,2π]
and λ ∈
[0,1]
. (As t goes from 0 to 2π the point γ
(t)
traces
the curve γ
([0,2π])
in the counter clockwise direction.) Suppose z ∈ D, the
inside of the simple closed curve and consider the curve δ
(t)
= z + re^{it} for
t ∈
[0,2π]
where r is chosen small enough that B
(z,r)
⊆ D. Then it happens that
n
(δ,z)
= n
(γ,z)
.
Proposition 50.7.17Under the above conditions,
n (δ,z) = n (γ,z)
and n
(δ,z)
= 1.
Proof: By changing the parameter, assume that
[a,b]
=
[0,2π]
. From Theorem
50.7.15 it suffices to assume also that γ is C^{1}. Define h_{λ}
(t)
≡ γ
(t)
+ λ
( it )
z + re − γ(t)
for λ ∈
[0,1]
. (This function is called a homotopy of the curves γ and δ.) Note that for
each λ ∈
This number is an integer and it is routine to verify that it is a continuous function of
λ. When λ = 0 it equals n
(γ,z)
and when λ = 1 it equals n
(δ,z)
. Therefore,
n
(δ,z)
= n
(γ,z)
. It only remains to compute n
(δ,z)
.
∫ 2π
n (δ,z) = -1- rieitdt = 1.
2πi 0 reit
This proves the proposition.
Now if γ was not one to one but caused the point, γ
(t)
to travel around γ^{∗} twice, you
could modify the above argument to have the parameter interval,
[0,4π]
and still
find n
(δ,z)
= n
(γ,z)
only this time, n
(δ,z)
= 2. Thus the winding number is
just what its name suggests. It measures the number of times the curve winds
around the point. One might ask why bother with the winding number if this is
all it does. The reason is that the notion of counting the number of times a
curve winds around a point is rather vague. The winding number is precise. It
is also the natural thing to consider in the general Cauchy integral formula
presented below. Consider a situation typified by the following picture in which Ω is
the open set between the dotted curves and γ_{j} are closed rectifiable curves in
Ω.