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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2015 • Question 9

Question

Math 6210 • Real Analysis Spring 2015 • Question 9

Let \(f\) be a holomorphic function defined on a domain \(\Omega \) in \(\mathbb C\). Show that \(f\) is either a polynomial, or there exists \(z\in \Omega \) such that \[ f^{(n)}(z)\ne 0 \] for all integers \(n\ge 0\).

Raw LaTeX

Let \(f\) be a holomorphic function defined on a domain \(\Omega\) in
\(\mathbb C\). Show that \(f\) is either a polynomial, or there exists
\(z\in\Omega\) such that
\[
f^{(n)}(z)\ne0
\]
for all integers \(n\ge0\).

Topic Tags

Holomorphic Functions and Cauchy-Riemann Equations Holomorphic Roots and Logarithms Zeros of Holomorphic Functions and the Identity Theorem