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Sample Midterm II Problems \hfill    November 6, 2000
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\begin{enumerate}

\item \begin{enumerate}

  \item Why does not the integral 
\[ \int_0^{\pi i} \sinh 5z dz \]
depend on the choice of contour connecting $0$ and $\pi i$?

  \item Evaluate this integral.
 \end{enumerate}

\item Evaluate
\[  \oint_C \bar{z}^2 dz \]
around the circle $|z-1| =1$ described counterclockwise.

\item Evaluate 
\[ \frac{1}{2\pi i} \oint_C \frac{e^{zt}}{(z^2+1)^2} dz \]
if $t > 0$ and $C$ is the circle \ (a) $|z| = 3$, \  (b) $|z| = 1/2$ \ described counterclockwise.

\item 
 \begin{enumerate}
 \item Show that 
\[\tan w = \frac{e^{2iw} -1}{i(e^{2iw}+1)}\].
 \item Show that the function $w = \arctan z$ that inverts the
function ${\displaystyle z = \tan w = \frac{\sin w}{\cos w}}$ can be
expressed as
\[ \arctan z = \frac{i}{2} \log \frac{i+z}{i-z}.  \]
 \end{enumerate}

\item
Evaluate the integral
\[\int_C e^{\overline{z}} dz      \]
over the unit circle $C: \; |z|=1$.

\end{enumerate}

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