Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 2010 • Question 6

Question

Math 6420 • PDE Fall 2010 • Question 6

For a function \(f(z)\) which is analytic within a simple, nonintersecting, closed curve \(C\) a corollary of the Cauchy integral formula implies that for any point \(z\) inside \(C\), \[ f'(z)=\frac {1}{2\pi i}\int _C\frac {f(\xi )}{(\xi -z)^2}\,d\xi . \] Use this to prove Liouville’s theorem that any function which is bounded and analytic in the entire complex plane must be a constant, and hence prove the fundamental theorem of algebra that an \(n\)th order polynomial \(P_n(z)\) has \(n\) roots (counting multiplicity) in the complex plane.

For a function \(f(z)\) which is analytic within a simple, nonintersecting, closed curve \(C\) a
corollary of the Cauchy integral formula implies that for any point \(z\) inside \(C\),
\[
  f'(z)=\frac{1}{2\pi i}\int_C\frac{f(\xi)}{(\xi-z)^2}\,d\xi.
\]
Use this to prove Liouville's theorem that any function which is bounded and analytic in
the entire complex plane must be a constant, and hence prove the fundamental theorem of
algebra that an \(n\)th order polynomial \(P_n(z)\) has \(n\) roots (counting multiplicity) in the
complex plane.

Topic Tags

Cauchy Integral Formula and Integral Theorems Entire Functions and Liouville's Theorem Fundamental Theorem of Algebra Linear Programming and Optimization