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\textbf{Math 425: Complex Variables}
\hfill\makebox[2.5in][r]{Name:\hrulefill}

\textbf{Quiz 5}\quad \textsl{April 23, 2004}

\bigskip

\begin{problem}{1.}%
In each case, write the principal part of the function at each of its
isolated singular points and determine whether those points are poles,
removable singular points, or essential singular points. Also determine the
residue of the function at each of the singular points.

(a) $\frac{\sin(z)}{z-\pi}$\hfill
(b) $z^2\cos(1/z)$\hfill 
(c) $\frac{z^2}{(z^2+1)^2}$
\hfill\ 
\end{problem}

\begin{problem}{2.}%
Use residues to calculate $\int_0^\infty\frac{x^2}{(x^2+1)^2}\,dx$.
\end{problem}

\newpage

\begin{problem}{3.}%
Use residues to calculate the principal value of
$\int_{-\infty}^\infty\frac{\cos(x)}{x^2+4x+5}\,dx$.
\end{problem}

\begin{problem}{4.}%
Use residues to calculate $\int_0^{2\pi}
\frac{1}{2+\cos(\theta)+\sin(\theta)}\,d\theta$.
\end{problem}

\end{document}
