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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Spring 2004 • Question 6

Question

Math 6420 • PDE Spring 2004 • Question 6

  1. Using Jordan’s lemma show that \[ \int _0^\infty \frac {\cos ax}{1+x^2}\,dx=\frac {\pi }{2}e^{-a}. \]
  2. Show that \[ \int _0^{2\pi }\frac {d\theta }{A+B\sin \theta }= \frac {2\pi }{\sqrt {A^2-B^2}} \] where \(A^2>B^2\), \(A>0\).

Raw LaTeX

\begin{enumerate}
\item[(a)] Using Jordan's lemma show that
\[
  \int_0^\infty \frac{\cos ax}{1+x^2}\,dx=\frac{\pi}{2}e^{-a}.
\]
\item[(b)] Show that
\[
  \int_0^{2\pi}\frac{d\theta}{A+B\sin\theta}=
  \frac{2\pi}{\sqrt{A^2-B^2}}
\]
where \(A^2>B^2\), \(A>0\).
\end{enumerate}

Topic Tags

Jordan's Lemma and Arc Estimates Linear Programming and Optimization Real Integral Evaluation by Complex Methods