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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2007 • Question 8

Question

Math 6210 • Real Analysis Fall 2007 • Question 8

Let \(f\) be an entire function such that for all \(x\in \mathbb R\), \(f(ix)\) and \(f(1+ix)\) are in \(\mathbb R\). Show that \(f\) is periodic with period \(2\). That is show that \(f(z)=f(z+2)\) for all \(z\in \mathbb C\).

Raw LaTeX

Let \(f\) be an entire function such that for all \(x\in\mathbb R\),
\(f(ix)\) and \(f(1+ix)\) are in \(\mathbb R\). Show that \(f\) is periodic with period
\(2\). That is show that \(f(z)=f(z+2)\) for all \(z\in\mathbb C\).

Topic Tags

Entire Functions and Liouville-Type Theorems