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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 1999 • Question 10

Question

Math 6420 • PDE Fall 1999 • Question 10

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.

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.

Topic Tags

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