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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 1995 • Question 7

Question

Math 6420 • PDE Fall 1995 • Question 7

  1. State and prove Jordan’s Lemma of complex function theory.
  2. Use Jordan’s Lemma and the residue theorem to prove the equation \[ \int _0^\infty \frac {\cos (x)}{x^2+a^2}\,dx=\frac {\pi e^{-a}}{2a} \] where \(a>0\).

Raw LaTeX

\begin{enumerate}
\item[(a)] State and prove Jordan's Lemma of complex function theory.
\item[(b)] Use Jordan's Lemma and the residue theorem to prove the equation
\[
  \int_0^\infty\frac{\cos(x)}{x^2+a^2}\,dx=\frac{\pi e^{-a}}{2a}
\]
where \(a>0\).
\end{enumerate}

Topic Tags

Contour Integration and Residue Calculus Isolated Singularities and Residues Jordan's Lemma and Arc Estimates Linear Programming and Optimization