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Sample Final Problems \hfill    December 8, 2000
\medskip

\begin{enumerate}

\item
 \begin{enumerate}
 \item If $z = 6 e^{\pi i/3}$, evaluate $|e^{iz}|$.
 \item Find all the values of $(-1)^i$, $(-1)^{\sqrt{2}}$.
 \end{enumerate}

\item 
 \begin{enumerate}
\item
Verify that the function $u(x,y) = \sin x \cosh y + x$ is harmonic.

\item Find the harmonic conjugate of $u(x,y)$.

\end{enumerate}

\item Evaluate the following integral over any simple closed contour
$C$ oriented positively around the point $z = -1/2$:
\[
\int_C \frac{z}{2z+1} dz.
\]

\item Find the maximum of $|f(z)|$ in $|z| \leq 1$ for the function
$f(z) = z^2 - 3 z + 2$.

\item
 \begin{enumerate}

\item
Show that a linear fractional transformation $T(z) =
\frac{az+b}{cz+d}$, where $ad-bc \ne 0$ and $c \ne 0$, always has
either one fixed point $\alpha$ or two distinct fixed points $\alpha$
and $\beta$.

\item 
If $T(z)$ has two fixed points $\alpha$ and $\beta$, show that $w =
T(z)$ satisfies the equation
\[
\frac{w-\alpha}{w - \beta} = k \frac{z-\alpha}{z-\beta},
\]
where 
\[
k = \frac{(-d/c) - \beta}{(-d/c)-\alpha}.
\]
[Hint: What are $w_1 = T(\alpha)$, $w_2 = T(-d/c)$, and $w_3 =
T(\beta)$?]
 \end{enumerate}

\item \begin{enumerate}

 \item Find the image of a vertical line $x = c \; (c > 0)$ under the
 transformation $w = 1/z$.

 \item Find the image of the common part of the domains $|z-1| > 1$
 and $|z-2| < 2$ under the mapping 
\[
w = \frac{i-e^{4\pi i (1/z - 1/4)}}{i+e^{4\pi i (1/z - 1/4)}}.
\]

 \end{enumerate}

\item

Find the images in the $w$-plane under the transformation $w = \sin z$
of the vertical lines $x = a $, $0 < a < \pi/2$, and the horizontal
line segments $y = b$, $0 < x < \pi/2$, sweeping out the vertical
strip $ 0 < \re z < \pi/2$.

\end{enumerate}

\end{document}
