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Mth 425 Midterm I  \hfill    October 6, 2000
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\begin{enumerate}

\item
Find the fifth roots of $-16 + 16 \sqrt{3} i$ and plot them in the
complex plane.
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\item
If $1= z_0$, $z_1$, \dots, $z_{n-1}$ are the $n$th roots of unity,
prove that
\[
(z-z_1) (z-z_2) \dots (z-z_{n-1}) = 1 + z + \dots + z^{n-1}.
\]

\newpage
\item   
Find the image of the region in the first quadrant that lies between
the hyperbolas $xy = 1/2$ and $xy = 4$ under the mapping $w =
z^2$. Sketch it in the plane.

\newpage
\item \begin{enumerate}

  \item Prove that the function $u = x e^x \cos y - y e^x \sin y$ is
  harmonic.  

\item Find the harmonic conjugate $v$ of $u$. 

\item
  Express $f(z) = u + i v$ in terms of $z$.
\end{enumerate}

\newpage
\item Prove that $z^2 \bar{z}$ is nowhere analytic.

\newpage
\item
Find all values of $z$ for which the equation
\[
e^z = -1 + i \sqrt{3}
\]
holds.

\newpage
\item
Show that
\[
\lim_{z\to 1 + i} \frac{z^2 - 2i}{z^2 - 2 z + 2} = 1 -i.
\]

\end{enumerate}

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