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\textbf{Math 425: Complex Variables}
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\textbf{Quiz 5}\quad \textsl{April 25, 2003}

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\begin{problem}{1.}%
Determine the types of the singular points of the following
functions:\newline
\emph{a)} $\sinh(z^4)/z^4$\qquad \emph{b)} $z^4\sinh(1/z^4)$\qquad
\emph{c)} $z^4/\sinh(z^4)$.
\end{problem}

\begin{problem}{2.}%
Use residues to show that $\int_0^\infty \frac{x^2\,dx}{(x^2+a^2)(x^2+b^2)}
=\frac{\pi}{2(a+b)}$.
\end{problem}

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

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\begin{problem}{4.}%
Use residues to compute $\int_0^{2\pi}\frac{d\theta}{1+\cos^2\theta}$.
\end{problem}

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\begin{problem}{5.}%
Let $f(z)$ be an entire function that satisfies $f(x) = x^2 + 1$ for $0 \le
x \le 1$. Find $f(2 + i)$.
\end{problem}

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