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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 1993 • Question 6

Question

Math 6210 • Real Analysis Fall 1993 • Question 6

(a) Let \(D\) be an open connected set in \(\mathbb C\), and \(f\) a holomorphic function on \(D\). Show that if either the real or imaginary part of \(f\) is constant, then \(f\) is constant.

(b) Let \(f\) be an entire function. Show that if either the real or imaginary part of \(f\) is a bounded function then \(f\) is a constant.

Raw LaTeX

(a) Let \(D\) be an open connected set in \(\mathbb C\), and \(f\) a
holomorphic function on \(D\). Show that if either the real or imaginary part of
\(f\) is constant, then \(f\) is constant.

(b) Let \(f\) be an entire function. Show that if either the real or imaginary
part of \(f\) is a bounded function then \(f\) is a constant.

Topic Tags

Entire Functions and Liouville-Type Theorems Holomorphic Functions and Cauchy-Riemann Equations Holomorphic Roots and Logarithms Riemann Mapping Theorem and Simply Connected Domains