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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Chapter 9<br />
Holomorphic Functions on Annuli<br />
Definition. Let z0 ∈ C, and let r,R ∈ [0,∞] be such that r < R. Then the annulus<br />
centered at z0 with inner radius r and outer radius R is defined as<br />
Ar,R(z0) := {z ∈ C : r < |z −z0|< R}.<br />
Theorem 9.1 (Cauchy’s Integral Theorem for Annuli). Let z0 ∈ C, let r,ρ,P,R ∈<br />
[0,∞] be such that r < ρ < P < R, and let f : Ar,R(z0) → C be holomorphic. Then<br />
we have � �<br />
f(ζ)dζ = f(ζ)dζ.<br />
∂BP(z0) ∂Bρ(z0)<br />
Pro<strong>of</strong>. The claim is equivalent to<br />
� �<br />
f(ζ)dζ +<br />
Consider<br />
∂BP(z0)<br />
z0<br />
r<br />
∂Bρ(z0) −<br />
57<br />
f(ζ)dζ = 0.<br />
ρ<br />
P<br />
R