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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Spring 2009 • Question 7

Question

Math 6420 • PDE Spring 2009 • Question 7

  1. Use contour integration to explicitly evaluate \[ f(a)=\int _0^\infty \frac {x\sin x\,dx}{x^2-a} \] for complex \(a\). For what values of complex \(a\) is this possible?
  2. Is the function \(f(a)\) an analytic function of \(a\)? Identify all singularities, branch points, branch cuts, etc.

Raw LaTeX

\begin{enumerate}
\item[(a)] Use contour integration to explicitly evaluate
\[
  f(a)=\int_0^\infty \frac{x\sin x\,dx}{x^2-a}
\]
for complex \(a\). For what values of complex \(a\) is this possible?

\item[(b)] Is the function \(f(a)\) an analytic function of \(a\)? Identify all singularities, branch
points, branch cuts, etc.
\end{enumerate}

Topic Tags

Branch Points, Branch Cuts, and Logarithms Contour Integration and Residue Calculus Isolated Singularities and Residues Linear Programming and Optimization