Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 1995 • Question 6

Question

Math 6420 • PDE Fall 1995 • Question 6

Liouville’s theorem states that a bounded entire function \(f(z)\) must be a constant.

  1. Prove Liouville’s theorem.
  2. Use Liouville’s theorem to prove the Fundamental Theorem of Algebra: every polynomial \(P_n(z)\) of degree \(n\ge 1\) has a root.

Raw LaTeX

Liouville's theorem states that a bounded entire function
\(f(z)\) must be a constant.
\begin{enumerate}
\item[(a)] Prove Liouville's theorem.
\item[(b)] Use Liouville's theorem to prove the Fundamental Theorem of Algebra: every
polynomial \(P_n(z)\) of degree \(n\ge 1\) has a root.
\end{enumerate}

Topic Tags

Entire Functions and Liouville's Theorem Fundamental Solutions Fundamental Theorem of Algebra Linear Programming and Optimization