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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Spring 2015 • Question 7

Question

Math 6210 • Real Analysis Spring 2015 • Question 7

Let \(f(z)=1+3z+z^n\).

(a) Show that \(f\) has exactly one root in the unit disk \(\{z:|z|<1\}\).

(b) Show that all roots of \(f\) lie in the disk \[ \left \{z:|z|\le \frac {3+\sqrt {13}}{2}\right \}. \]

Raw LaTeX

Let \(f(z)=1+3z+z^n\).

(a) Show that \(f\) has exactly one root in the unit disk \(\{z:|z|<1\}\).

(b) Show that all roots of \(f\) lie in the disk
\[
\left\{z:|z|\le\frac{3+\sqrt{13}}{2}\right\}.
\]

Topic Tags

Argument Principle and Rouche's Theorem