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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2019 • Question 9

Question

Math 6210 • Real Analysis Fall 2019 • Question 9

Prove that if \(f\) is entire and sends the real line to itself, then \[ f(\bar z)=\overline {f(z)} \] for all \(z\in \mathbb C\). In words, applying \(f\) to the conjugate input gives the complex conjugate of the original output.

Raw LaTeX

Prove that if \(f\) is entire and sends the real line to itself,
then
\[
f(\bar z)=\overline{f(z)}
\]
for all \(z\in\mathbb C\). In words, applying \(f\) to the conjugate input gives
the complex conjugate of the original output.

Topic Tags

Entire Functions and Liouville-Type Theorems