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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Theorem 7.5 (Biholomorphisms <strong>of</strong> D). Let f: D → D be biholomorphic. Then there<br />
exist w ∈ D and c ∈ ∂D with f(z) = cφw(z) for z ∈ D.<br />
Pro<strong>of</strong>. Set w := f −1 (0). Then f ◦φw: D → D is biholomorphic with (f ◦φw)(0) = 0.<br />
By Corollary 7.4.1, there exists c ∈ C with |c|= 1 such that f(φw(z)) = cz for z ∈ D,<br />
so that<br />
f(z) = f(φw(φw(z))) = cφw(z)<br />
for z ∈ D.<br />
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