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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2008 • Question 10

Question

Math 6210 • Real Analysis Fall 2008 • Question 10

Let \(f\) be a function holomorphic on \(D\) and continuous on \(\overline {D}\). Assume that \(|f(z)|=1\) whenever \(|z|=1\). Show that \(f\) can be extended to a meromorphic function on the whole \(\mathbb C\), with at most finitely many poles.

(Hint: Starting with the Schwarz reflection principle for the upper half plane, deduce that an appropriate reflection continuation that can be used in this setting is \(z\mapsto \frac {1}{f(1/\bar z)}\).)

Raw LaTeX

Let \(f\) be a function holomorphic on \(D\) and continuous on
\(\overline{D}\). Assume that \(|f(z)|=1\) whenever \(|z|=1\). Show that \(f\) can be
extended to a meromorphic function on the whole \(\mathbb C\), with at most
finitely many poles.

(\textit{Hint:} Starting with the Schwarz reflection principle for the upper
half plane, deduce that an appropriate reflection continuation that can be used
in this setting is \(z\mapsto \frac{1}{f(1/\bar z)}\).)

Topic Tags

Argument Principle and Rouche's Theorem Holomorphic Functions and Cauchy-Riemann Equations Isolated Singularities and Residues Meromorphic Functions and Rational Functions