Math 411: Honours Complex Variables - University of Alberta
Math 411: Honours Complex Variables - University of Alberta
Math 411: Honours Complex Variables - University of Alberta
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66 CHAPTER 10. THE WINDING NUMBER OF A CURVE<br />
so that<br />
and thus<br />
exp(Lj(γ(tj))−Lj+1(γ(tj))) = 1<br />
Lj(γ(tj))−Lj+1(γ(tj)) ∈ 2πiZ.<br />
On differentiating e Lj(w) = w−z, we find that e Lj(w) L ′ j (w) = 1. Thus<br />
this allows us to express<br />
�<br />
γ<br />
1<br />
dζ =<br />
ζ −z<br />
�<br />
We thus see that<br />
=<br />
=<br />
n�<br />
�<br />
j=1<br />
L ′ 1<br />
j (w) =<br />
w−z<br />
γ| [tj−1 ,t j ]<br />
1<br />
ζ −z dζ<br />
n�<br />
[Lj(γ(tj))−Lj(γ(tj−1))]<br />
j=1<br />
for w ∈ Dj;<br />
n� �n−1<br />
Lj(γ(tj))− Lj+1(γ(tj))<br />
j=1<br />
j=0<br />
�n−1<br />
= Ln(γ(tn))−L1(γ(t0))+ [Lj(γ(tj))−Lj+1(γ(tj))].<br />
j=1<br />
�n−1<br />
= Ln(γ(tn))−Ln+1(γ(tn))+ [Lj(γ(tj))−Lj+1(γ(tj))].<br />
γ<br />
=<br />
j=1<br />
n�<br />
[Lj(γ(tj))−Lj+1(γ(tj))].<br />
j=1<br />
1<br />
dζ ∈ 2πiZ.<br />
ζ −z<br />
Definition. Let γ be a closed curve in C. We define the interior and exterior <strong>of</strong> γ<br />
to be<br />
and<br />
intγ := {z ∈ C\{γ} : ν(γ,z) �= 0}<br />
extγ := {z ∈ C\{γ} : ν(γ,z) = 0}.