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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2019 • Question 7

Question

Math 6210 • Real Analysis Fall 2019 • Question 7

The following statement is false: If \(\Omega \subset \mathbb C\) is any simply connected set and \(x\in \Omega \), then there exists a unique \(r\) and biholomorphism \(f:\Omega \to B(0,r)\) so that \(f(x)=0\) and \(f'(x)=1\). Fix the statement and provide a proof for the correct statement.

Note: Simply connected implies the set is open and connected.

Raw LaTeX

The following statement is false: If \(\Omega\subset\mathbb C\) is
any simply connected set and \(x\in\Omega\), then there exists a unique \(r\) and
biholomorphism \(f:\Omega\to B(0,r)\) so that \(f(x)=0\) and \(f'(x)=1\). Fix the
statement and provide a proof for the correct statement.

\textbf{Note:} Simply connected implies the set is open and connected.

Topic Tags

Automorphism Groups and Biholomorphisms Riemann Mapping Theorem and Simply Connected Domains