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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 2019 • Question 7

Question

Math 6420 • PDE Fall 2019 • Question 7

Let \(f(z)\) be an entire function such that there exists a constant \(M\), an \(R>0\) and an integer \(n\ge 1\) with \(\lvert f(z)\rvert \le M\lvert z\rvert ^n\) for \(\lvert z\rvert >R\). Show that \(f(z)\) is a polynomial of degree at most \(n\).

Raw LaTeX

Let \(f(z)\) be an entire function such that there exists a constant \(M\), an \(R>0\) and an
integer \(n\ge1\) with \(\lvert f(z)\rvert\le M\lvert z\rvert^n\) for \(\lvert z\rvert>R\).
Show that \(f(z)\) is a polynomial of degree at most \(n\).

Topic Tags

Entire Functions and Liouville's Theorem Linear Programming and Optimization Polynomial Interpolation and Error Formulas