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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6220 • Complex Analysis

Summer 2022 • Question 1

Question

Math 6220 • Complex Analysis Summer 2022 • Question 1

Let \(\Omega \) be a simply connected domain in \(\mathbb {C}\). Show that if a holomorphic function \(f:\Omega \to \mathbb {C}\) has finitely many zeros, all of even order, then \(f\) has a holomorphic square root in \(\Omega \), i.e. there is a holomorphic function \(g:\Omega \to \mathbb {C}\) so that \(f(z)=g(z)^2\).

Raw LaTeX

Let \(\Omega\) be a simply connected domain in \(\mathbb{C}\). Show that if a holomorphic
function \(f:\Omega\to\mathbb{C}\) has finitely many zeros, all of even order, then \(f\)
has a holomorphic square root in \(\Omega\), i.e. there is a holomorphic function
\(g:\Omega\to\mathbb{C}\) so that \(f(z)=g(z)^2\).

Topic Tags

Holomorphic Functions and Cauchy-Riemann Equations Holomorphic Roots and Logarithms Riemann Mapping Theorem and Simply Connected Domains Zeros of Holomorphic Functions and the Identity Theorem