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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 2003 • Question 6

Question

Math 6420 • PDE Fall 2003 • Question 6

  1. Evaluate the following real integrals by contour integration: \[ \int _0^{2\pi }\frac {d\theta }{(5-3\sin \theta )^2}. \]
  2. Let \[ e^{t/2(z-1/z)}=\sum _{n=-\infty }^{\infty }J_n(t)z^n. \] Show from the definition of Laurent series and properties of complex integration that the Bessel function \[ J_n(t)=\frac {1}{\pi }\int _0^\pi \cos (n\theta -t\sin \theta )\,d\theta . \]

Raw LaTeX

\begin{enumerate}
\item[(a)] Evaluate the following real integrals by contour integration:
\[
  \int_0^{2\pi}\frac{d\theta}{(5-3\sin\theta)^2}.
\]
\item[(b)] Let
\[
  e^{t/2(z-1/z)}=\sum_{n=-\infty}^{\infty}J_n(t)z^n.
\]
Show from the definition of Laurent series and properties of complex integration that the
Bessel function
\[
  J_n(t)=\frac{1}{\pi}\int_0^\pi \cos(n\theta-t\sin\theta)\,d\theta.
\]
\end{enumerate}

Topic Tags

Contour Integration and Residue Calculus Laurent Series and Annuli Linear Programming and Optimization Real Integral Evaluation by Complex Methods