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\textbf{Math 425: Complex Variables}
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\textbf{Quiz 4}\quad \textsl{April 4, 2003}

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\begin{problem}{1.}%
Find an analytic function $f(z)$ whose Taylor series is $\sum_{n=1}^\infty
\frac{n}{n+1}\,z^n$.
\end{problem}

\begin{problem}{2.}%
Determine the radius of convergence of $\sum_{n=1}^\infty 
\frac{n! - i^n}{n^n}\,z^n$.
\end{problem}

\begin{problem}{3.}%
Use multiplication and division of series to determine the first three
non-zero terms in the Taylor series expansion of
$\frac{e^z\log(1-z)}{\cos(z)}$ at $z=0$.
\end{problem}

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\begin{problem}{4.}%
Find the residue of $f(z) = \frac{\cosh(z)}{z(z^2+1)^2}$ at each of its
isolated singular points.
\end{problem}

\begin{problem}{5.}%
Calculate $\int_{\CC} \frac{\cosh(z)}{z(z^2+1)^2}\,dz$ where $\CC$ is the
circle $|z| = 2$ oriented counter-clockwise.  \end{problem}

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