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Sample Midterm Problems \hfill    September 27, 2000
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\begin{enumerate}

\item
Use De Moivre's formula to write $\cos 4\theta$ through $\cos \theta$
and $\sin \theta$.

\item
Find all cube roots of $-i$. Sketch them on the complex plane.

\item
Using the quadratic formula for the roots of a polynomial $a z^2 + bz
+ c$ with \emph{real} coefficients $a$, $b$, and $c$, show that when
$b^2 - 4ac < 0$, the polynomial has two complex conjugate roots.

\item
Let $\omega$ be any $n$th root of unity other than 1. Prove the
identity
\[
1 + \omega + \omega^2 + \dots + \omega^{n-1} = 0.
\]

\item
A square $S$ in the $z$ plane has vertices at $(0,0),\; (1,0),
\;(1,1),\; (0,1)$. Determine the region in the $w$ plane into which
$S$ is mapped under the transformation $w = z^2$. Sketch it on the
plane.

\item Prove that $z + \bar{z}^3$ is nowhere analytic.

\item
Find an analytic function $f(z) = u(x,y) + i v(x,y)$ so that $u(x,y) =
\sin y \sinh x$.

\item
Verify that
\[
\lim_{z \to 0} \frac{e^z - 1}{z} = 1.
\]

\end{enumerate}

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