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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6220 • Complex Analysis

Fall 2022 • Question 1

Question

Math 6220 • Complex Analysis Fall 2022 • Question 1

Let \(\Omega \) be a simply connected domain in \(\mathbb {C}\). Suppose that \(f:\Omega \to \mathbb {C}\) is holomorphic and has finitely many zeros in \(\Omega \). What is the supremum of \(m\in \mathbb {N}\) so that \(f(z)=g(z)^m\) for some \(g\) holomorphic in \(\Omega \)? Identify this value explicitly in terms of the orders of the zeros of \(f\) and prove that it is correct.

Raw LaTeX

Let \(\Omega\) be a simply connected domain in \(\mathbb{C}\). Suppose that \(f:\Omega\to\mathbb{C}\) is
holomorphic and has finitely many zeros in \(\Omega\). What is the supremum of
\(m\in\mathbb{N}\) so that \(f(z)=g(z)^m\) for some \(g\) holomorphic in \(\Omega\)? Identify this
value explicitly in terms of the orders of the zeros of \(f\) and prove that it is
correct.

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