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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2009 • Question 9

Question

Math 6210 • Real Analysis Spring 2009 • Question 9

Let \(Q\) be a rectangle in \(\mathbb C\) and \(f:\mathbb C\to \mathbb C\) a holomorphic map. Assume that there exists a \(z\in Q\) such that \(f(z)=z\) and \(f'(z)=1\). Show that \(f\) is the identity map.

Raw LaTeX

Let \(Q\) be a rectangle in \(\mathbb C\) and
\(f:\mathbb C\to\mathbb C\) a holomorphic map. Assume that there exists a
\(z\in Q\) such that \(f(z)=z\) and \(f'(z)=1\). Show that \(f\) is the identity map.

Topic Tags

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