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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Spring 2014 • Question 6

Question

Math 6420 • PDE Spring 2014 • Question 6

The function \(f(z)\) is analytic in the extended complex plane except for two points \(z=0\) and \(z=2\), where it has simple poles. It is known that \[ \oint _{|z|=1} f(z)\,dz=1, \qquad \oint _{|z|=3} f(z)\,dz=0. \] Find \(f(z)\) (unique up to an arbitrary additive constant).

Raw LaTeX

The function \(f(z)\) is analytic in the extended complex plane except for two points \(z=0\)
and \(z=2\), where it has simple poles. It is known that
\[
  \oint_{|z|=1} f(z)\,dz=1,
  \qquad
  \oint_{|z|=3} f(z)\,dz=0.
\]
Find \(f(z)\) (unique up to an arbitrary additive constant).

Topic Tags

Isolated Singularities and Residues Linear Programming and Optimization