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Name \hrulefill \\

Mth 425 Final  \hfill    December 14, 2000
\medskip

Read the problems carefully. Show all work, refer to results you use
by name, such as the Cauchy Theorem, or by content, such as ``the
theorem that says the integral over a simple closed contour of a
function analytic on and inside the contour is zero''.
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\begin{enumerate}

\item
Find all complex values of $\sqrt[3]{1-i\sqrt{3}}$.

\newpage
\item
Find a harmonic conjugate for $u(x,y) = \cosh x \cos y - x^2 + y^2$.

\newpage
\item
Find the principal value of $\arctan (1+2i)$. [Hint: you do not have
to know a formula for $\arctan$ to solve this problem. However, you
need to know a formula for $\tan z$.]

\newpage
\item
Evaluate the following integral over the circle $|z-1| = 1/2$ oriented
counterclockwise:
\[
\int_C \frac{e^z}{z(1-z)^3} dz.
\]

\newpage
\item
Find all points of the unit disk $|z| \le 1$ at which the function
$e^z$ has a maximal modulus.

\newpage
\item
Find a linear fractional transformation $w = \frac{az+b}{cz+d}$,
$ad-bc \ne 0$, which maps the circle $|z+1| = 1$ onto the $u$ axis on
the $w$ plane. [Hint: pick three points on the circle.]

\newpage
\item
Find the image of the disk $|z| < 1$ under the mapping
\[
w = \left(\frac{1+z}{1-z} \right)^2.
\]

\end{enumerate}

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