Math 411: Honours Complex Variables - University of Alberta

66 CHAPTER 10. THE WINDING NUMBER OF A CURVE
so that
and thus
exp(Lj(γ(tj))−Lj+1(γ(tj))) = 1
Lj(γ(tj))−Lj+1(γ(tj)) ∈ 2πiZ.
On differentiating e Lj(w) = w−z, we find that e Lj(w) L ′ j (w) = 1. Thus
this allows us to express
�
γ
1
dζ =
ζ −z
�
We thus see that
=
=
n�
�
j=1
L ′ 1
j (w) =
w−z
γ| [tj−1 ,t j ]
1
ζ −z dζ
n�
[Lj(γ(tj))−Lj(γ(tj−1))]
j=1
for w ∈ Dj;
n� �n−1
Lj(γ(tj))− Lj+1(γ(tj))
j=1
j=0
�n−1
= Ln(γ(tn))−L1(γ(t0))+ [Lj(γ(tj))−Lj+1(γ(tj))].
j=1
�n−1
= Ln(γ(tn))−Ln+1(γ(tn))+ [Lj(γ(tj))−Lj+1(γ(tj))].
γ
=
j=1
n�
[Lj(γ(tj))−Lj+1(γ(tj))].
j=1
1
dζ ∈ 2πiZ.
ζ −z
Definition. Let γ be a closed curve in C. We define the interior and exterior of γ
to be
and
intγ := {z ∈ C\{γ} : ν(γ,z) �= 0}
extγ := {z ∈ C\{γ} : ν(γ,z) = 0}.