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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58 CHAPTER 9. HOLOMORPHIC FUNCTIONS ON ANNULI<br />
Split∂BP(z0)and∂Bρ(z0) − int<strong>of</strong>initelymanyarcsegmentsα1,...,αn andβ1,...,βn,<br />
respectively, andconnectthemwithlinesegmentsγ1,...,γn asshownbelowforn = 3:<br />
We thus obtain<br />
�<br />
∂BP(z0)<br />
α3<br />
�<br />
f(ζ)dζ +<br />
γ − 3<br />
γ3<br />
β3<br />
∂Bρ(z0) −<br />
γ1<br />
γ − 1<br />
β2<br />
α2<br />
f(ζ)dζ =<br />
=<br />
n�<br />
�<br />
j=1<br />
n�<br />
�<br />
j=1<br />
β1<br />
αj<br />
γ − 2<br />
α1<br />
γ2<br />
f(ζ)dζ +<br />
n�<br />
�<br />
j=1<br />
βj<br />
αj⊕γ −<br />
j ⊕βj⊕γ (j+1)mod n<br />
f(ζ)dζ<br />
f(ζ)dζ<br />
Bymakingthearcsegmentsα1,...,αn andβ1,...,βn sufficiently small, wecanensure<br />
that each <strong>of</strong> the closed curves α1 ⊕γ − 1 ⊕β1 ⊕γ2,...,αn−1 ⊕γ − n−1 ⊕βn−1 ⊕γn,αn ⊕<br />
γ − n ⊕βn ⊕γ1 lies inside a star-shaped open subset <strong>of</strong> Ar,R(z0):