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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 1998 • Question 8

Question

Math 6420 • PDE Fall 1998 • Question 8
Liouville's theorem states that a function analytic and bounded in the complex plane is necessarily a constant. Prove this theorem. Using it prove the Fundamental theorem of algebra: every polynomial \(P(z)\) which is not a constant has at least one complex root. **Applied Math Qualifying Exam Part C** Solve two of the four problems for full credit on Part C.

Raw LaTeX

Liouville's theorem states that a function analytic and
bounded in the complex plane is necessarily a constant.

Prove this theorem. Using it prove the Fundamental theorem of algebra: every polynomial
\(P(z)\) which is not a constant has at least one complex root.
\end{enumerate}

\examheading{Applied Math Qualifying Exam Part C}

Solve two of the four problems for full credit on Part C.

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Topic Tags

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