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\begin{document}
{\bf\large Math 425}\hfill {\bf\large Fall 1997}

\begin{center}
{\bf Review for Exam~3}
\end{center}

{\bf 1.}\hfill\begin{minipage}[t]{18cm}
Find all singularities and classify them as removable singularities, poles,
or essential
singularities.
Find also the order of each of the poles and the value of the function at each
removable singularity.
\\[1ex]
(a) ${\Ds f(z)=\frac{e^{2\pi z}-1}{(z^2+1)^2 z}}$
\\[1ex]
(b) ${\Ds f(z)=\frac{z}{\cos z-1}}$
\end{minipage}

{\bf 2.}\hfill\begin{minipage}[t]{18cm}
Find the residue of the following function at all  singularities
\[
     f(z)=\frac{\cos z -1}{(z^2+1)z^3}
\]
\end{minipage}

{\bf 3.} \hfill\begin{minipage}[t]{18cm}
Find the principal part of the Laurent-series and the residue of the
following function
at the indicated point.
\\[1ex]
(a) ${\Ds f(z)=\frac{\cos z}{(z^2-\pi^2)^2}}\ $  at $z_0=\pi$
\\[1ex]
(b) ${\Ds f(z)=\frac{z^3+6z^2+13z+11}{(z+2)^3}}\ $ at $z_0=-2$
\end{minipage}

{\bf 4.}\hfill\begin{minipage}[t]{18cm}
Calculate the following real integrals.
\\[1ex]
(a) ${\Ds \int_{-\infty}^{\infty}\frac{1}{x^4+5x^2+4}dx}$
\\[1ex]
(b) ${\Ds \int_{-\infty}^{\infty}\frac{\cos x}{(x^2+1)^2}dx}$
\end{minipage}

{\bf 5.}\hfill\begin{minipage}[t]{18cm}
Calculate the following integrals.
\\[1ex]
(a) ${\Ds \int_{|z|=1}\frac{\cos z-1}{z(z-2)^3}dz}$
\\[1ex]
(b) ${\Ds \int_{|z-\pi|=1}\frac{\sin z}{(z-\pi)^7}dz}$
\\[1ex]
(c) ${\Ds \int_{|z-2|=2}\frac{e^{2iz}}{(z^2-1)(z-2)}dz}$
\end{minipage}
\end{document}
