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\begin{document}
{\bf Name:} \underline{\mbox{\hspace*{7cm}}}
\hfill{\bf Instructor:} Heide Gluesing-Luerssen

\begin{center}
{\bf Mathematics 425: Complex Variables }
\\
{\bf Fall Semester 1997}
\\
{\bf Exam 1}
\\
{\bf September 19, 1997}
\end{center}
\vspace*{.4cm}

\begin{tabular}{p{18cm}}\hline
This
\rule[0cm]{0cm}{.8cm}
Examination contains 7 problems on 7 sheets of paper including the front cover
and one extra sheet on the back.
Do all your work in this booklet and show your computations.
Calculators, books, and notes are not allowed.
\\[1.7ex]
\hline
\end{tabular}

\begin{center} {\bf Scores}

\begin{tabular}{|c|c|c|}\hline
Question & Possible\quad &\ Actual\quad\rule[-.5cm]{0cm}{1.2cm} \\ \hline\hline
  1      &   10     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
  2      &   10     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
  3      &   15     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
  4      &   15     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
  5      &   15     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
  6      &   20     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
  7      &   15     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline\hline
Total    & 100\mbox{\  \ } & \rule[-.5cm]{0cm}{1.2cm} \\ \hline
\end{tabular}
\end{center}

\vspace*{.5cm}
{\bf Sign the pledge:}
\\
``On my honor, I have neither given nor received unauthorized aid on this
Exam.'':

{\bf Signature:} \underline{\mbox{\hspace*{7cm}}}

\vspace*{.5cm}
\begin{center}     {\bf GOOD LUCK}\end{center}

\newpage
{\bf 1.} \begin{minipage}[t]{17.5cm} (10 points)
Compute the following expressions.
\\[2ex]
(a) ${\Ds \frac{(3+i)(1-3i)}{1+i}}$.
\vspace*{5.8cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{8cm}}}
\\[2ex]
(b) ${\Ds \Big(\frac{-1-i\sqrt{3}}{2}\Big)^3}$.
\vspace*{5.8cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{8cm}}}
\end{minipage}
\\[3.5ex]
{\bf 2.} \begin{minipage}[t]{17.5cm} (10 points)
Find the number ${\Ds
\Big(-\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}i\Big)^{100}}$.
\vspace*{7cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{8cm}}}
\end{minipage}

\newpage
{\bf 3.} \begin{minipage}[t]{17.5cm} (15 points)
Find all 3rd roots of -8 and write them in standard form $a+bi$.
\vspace*{8cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{8cm}}}
\end{minipage}
\\[2ex]
{\bf 4.} \begin{minipage}[t]{17.5cm} (15 points)
Find the limit of each function at the given point, or explain why it does not
exist.
\\[2ex]
(a) $f(z)= {\rm Arg} (z)$ at the point $z_0=-1$.
\vspace*{5.8cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{8cm}}}
\\[2ex]
(b) ${\Ds g(z)=\frac{z^3-8i}{z+2i}}$ for $z\in\C,\;z\not=-2i$ at the point
$z_0=-2i$.
\vspace*{5.8cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{8cm}}}
\end{minipage}

\newpage

{\bf 5.} \begin{minipage}[t]{17.5cm} (15 points)
Is the set described in the following open?
Is the interior of the set connected?
Show the set and sketch your ideas for the answers on the diagram.
\\[2ex]
(a) $\{z\in\C\,|\, -1<\re((1+i)z)<1\}$.

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\\[6ex]
{\bf Answer:} \underline{\mbox{\hspace*{8cm}}}
\\[6ex]
(b) $\{z\in\C\,|\, 2\leq |z| < 4\}$.

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\\[6ex]
{\bf Answer:} \underline{\mbox{\hspace*{8cm}}}
\end{minipage}

\newpage
{\bf 6.} \begin{minipage}[t]{17.5cm} (20 points)
Compute the value(s) of the following expressions.
\\[2ex]
(a) $e^{-i\frac{\pi}{2}}$
\vspace*{4.5cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{8cm}}}
\\[2ex]
(b) $\log(-1-i)$
\vspace*{4.5cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{8cm}}}
\\[2ex]
(c) $i^i$
\vspace*{4.5cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{8cm}}}
\\[2ex]
(d) $\cos\pi i$
\vspace*{4.5cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{8cm}}}
\end{minipage}

\newpage
{\bf 7.} \begin{minipage}[t]{17.5cm} (15 points)
Prove the identity $\cos 2z = (\cos z)^2-(\sin z)^2$ for every complex number
$z$.
\end{minipage}

\newpage
\mbox{}










\end{document}
