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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2008 • Question 6

Question

Math 6210 • Real Analysis Spring 2008 • Question 6

Let \(Q\) be a square in \(\mathbb C\) and \(f:Q\to \mathbb C\) a continuous map that is holomorphic on the interior of \(Q\). Also assume that if \(z\in \partial Q\) then \(|f(z)|=1\). Show that \(f\) extends to a holomorphic map on a neighborhood of \(Q\).

Raw LaTeX

Let \(Q\) be a square in \(\mathbb C\) and
\(f:Q\to\mathbb C\) a continuous map that is holomorphic on the interior of
\(Q\). Also assume that if \(z\in\partial Q\) then \(|f(z)|=1\). Show that \(f\)
extends to a holomorphic map on a neighborhood of \(Q\).

Topic Tags

Holomorphic Functions and Cauchy-Riemann Equations Holomorphic Roots and Logarithms Zeros of Holomorphic Functions and the Identity Theorem