%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\documentclass[12pt]{article}
\usepackage{amssymb}
\topmargin-3cm
\textheight 25cm
\textwidth18.5cm
\oddsidemargin-.7cm
\evensidemargin-.7cm
%\pagestyle{empty}
\parindent0cm
\parskip 2.5ex
\newcommand{\re}{\rm Re}
\renewcommand{\bar}{\overline}
\newcommand{\im}{\rm Im}
\newcommand{\C}{\mathbb C}
\newcommand{\Ds}{\displaystyle}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
{\bf Name:} \underline{\mbox{\hspace*{7cm}}}
\hfill{\bf Instructor:} Bei Hu

\begin{center}
{\bf Mathematics 425: Complex Variables }
\\
{\bf Spring Semester 1999}
\\
{\bf Exam 1}
\\
{\bf February 5, 1999}
\end{center}
\vspace*{.4cm}

\begin{tabular}{p{18cm}}\hline
This
\rule[0cm]{0cm}{.8cm}
Examination contains 8 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      &   10     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
  7      &   10     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
  8      &   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)
(a) For $\zeta=4+3i$, $w=2-i$, compute
${\Ds \frac{\bar w \zeta}{w}}$.
\vspace*{5.8cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{8cm}}}
\\[2ex]
(b) Compute ${\Ds \Big(\frac{1+i}{\sqrt2}\Big)^4}$.
\vspace*{5.8cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{8cm}}}
\end{minipage}
\\[3.5ex]
{\bf 2.} \begin{minipage}[t]{17.5cm} (10 points)
Determine whether the points $1+i$, $4+2i$, $3+6i$ constitute
the vertices of a right triangle.
\vspace*{7cm}

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

\newpage
{\bf 3.} \begin{minipage}[t]{17.5cm} (15 points)
Find all 5 roots of $z^5=2-2i$. 
\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, or explain why it does not
exist.
\\[2ex]
(a) $f(z)= {\rm Log} (z)$, $z_0=-1$. Find
$\Ds \lim_{z\to z_0} f(z)$.
\vspace*{5.8cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{8cm}}}
\\[2ex]
(b) 
 Find $\Ds \lim_{n\to\infty}  \Big(\frac{1+3i}{4}\Big)^n$.
\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 {\bf open?} {\bf closed?}
Is the {\bf interior} of the set {\bf connected?}
Show the set and sketch your ideas for the answers on the diagram.
\\[2ex]
(a) $\{z\in\C\,|\, 1<|z|<3\}$.

\unitlength1cm
\hspace*{9cm}\begin{picture}(8.5,8.5)
\put(-.5,4){\vector(1,0){9}}
\put(4,-.5){\vector(0,1){9}}
\put(0,3.9){\line(0,1){.2}}
\put(1,3.9){\line(0,1){.2}}
\put(2,3.9){\line(0,1){.2}}
\put(3,3.9){\line(0,1){.2}}
\put(5,3.9){\line(0,1){.2}}
\put(6,3.9){\line(0,1){.2}}
\put(7,3.9){\line(0,1){.2}}
\put(8,3.9){\line(0,1){.2}}
\put(5.1,3.5){$1$}
\put(3.9,5){\line(1,0){.2}}
\put(3.9,0){\line(1,0){.2}}
\put(3.9,1){\line(1,0){.2}}
\put(3.9,2){\line(1,0){.2}}
\put(3.9,3){\line(1,0){.2}}
\put(3.9,6){\line(1,0){.2}}
\put(3.9,7){\line(1,0){.2}}
\put(3.9,8){\line(1,0){.2}}
\put(3.5,4.8){$i$}
\end{picture}
\\[6ex]
{\bf Answer:} \underline{\mbox{\hspace*{8cm}}}
\\[6ex]
(b) $\{z=r e^{i\theta}\,|\, 0\le r\le 3, \pi/2\le\theta\le 2\pi\}$.

\unitlength1cm
\hspace*{9cm}\begin{picture}(8.5,8.5)
\put(-.5,4){\vector(1,0){9}}
\put(4,-.5){\vector(0,1){9}}
\put(0,3.9){\line(0,1){.2}}
\put(1,3.9){\line(0,1){.2}}
\put(2,3.9){\line(0,1){.2}}
\put(3,3.9){\line(0,1){.2}}
\put(5,3.9){\line(0,1){.2}}
\put(6,3.9){\line(0,1){.2}}
\put(7,3.9){\line(0,1){.2}}
\put(8,3.9){\line(0,1){.2}}
\put(5.1,3.5){$1$}
\put(3.9,5){\line(1,0){.2}}
\put(3.9,0){\line(1,0){.2}}
\put(3.9,1){\line(1,0){.2}}
\put(3.9,2){\line(1,0){.2}}
\put(3.9,3){\line(1,0){.2}}
\put(3.9,6){\line(1,0){.2}}
\put(3.9,7){\line(1,0){.2}}
\put(3.9,8){\line(1,0){.2}}
\put(3.5,4.8){$i$}
\end{picture}
\\[6ex]
{\bf Answer:} \underline{\mbox{\hspace*{8cm}}}
\end{minipage}

\newpage
{\bf 6.} \begin{minipage}[t]{17.5cm} (10 points)
Compute the value(s) of the following expressions, and
write in the format $a+bi$.
\\[2ex]
(a) $e^{3\pi i/4}$
\vspace*{4.0cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{8cm}}}
\\[2ex]
(b) $\log(\sqrt3-i)$
\vspace*{4.0cm}

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

{\bf 7.} \begin{minipage}[t]{17.5cm} (10 points) Determine
whether the following series converge, indicating the test
you used.
\\[2ex]
(a) $\Ds \sum_{n=1}^\infty i^n$
\vspace*{4.0cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{8cm}}}
\\[2ex]
(b) $\Ds \sum_{n=1}^\infty \frac{(3+4i)^n}{n!} $
\vspace*{4.0cm}

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

\newpage
{\bf 8.} \begin{minipage}[t]{17.5cm} (15 points)

(a) Sketch the set $A=\{z=x+yi\;|\; \pi/4<y<\pi/2\}$ on $z$-plane.

(b) Sketch the image of the set $A$ under the map
$w=e^z$ in $w$-plane. This set is denoted by $B$.

(c) Sketch the image of the set $B$ under the map
$\zeta = w^3$ in $\zeta$-plane.  Give your explanation
in the spaces provided.  



\unitlength.75cm
\hspace*{9cm}\begin{picture}(8.5,8.5)
\put(-.5,4){\vector(1,0){9}}
\put(4,-.5){\vector(0,1){9}}
\put(0,3.9){\line(0,1){.2}}
\put(1,3.9){\line(0,1){.2}}
\put(2,3.9){\line(0,1){.2}}
\put(3,3.9){\line(0,1){.2}}
\put(5,3.9){\line(0,1){.2}}
\put(6,3.9){\line(0,1){.2}}
\put(7,3.9){\line(0,1){.2}}
\put(8,3.9){\line(0,1){.2}}
\put(5.1,3.5){$1$}
\put(3.9,5){\line(1,0){.2}}
\put(3.9,0){\line(1,0){.2}}
\put(3.9,1){\line(1,0){.2}}
\put(3.9,2){\line(1,0){.2}}
\put(3.9,3){\line(1,0){.2}}
\put(3.9,6){\line(1,0){.2}}
\put(3.9,7){\line(1,0){.2}}
\put(3.9,8){\line(1,0){.2}}
\put(3.5,4.8){$i$}
\end{picture}

\vspace*{3em}

\hspace*{9cm}\begin{picture}(8.5,8.5)
\put(-.5,4){\vector(1,0){9}}
\put(4,-.5){\vector(0,1){9}}
\put(0,3.9){\line(0,1){.2}}
\put(1,3.9){\line(0,1){.2}}
\put(2,3.9){\line(0,1){.2}}
\put(3,3.9){\line(0,1){.2}}
\put(5,3.9){\line(0,1){.2}}
\put(6,3.9){\line(0,1){.2}}
\put(7,3.9){\line(0,1){.2}}
\put(8,3.9){\line(0,1){.2}}
\put(5.1,3.5){$1$}
\put(3.9,5){\line(1,0){.2}}
\put(3.9,0){\line(1,0){.2}}
\put(3.9,1){\line(1,0){.2}}
\put(3.9,2){\line(1,0){.2}}
\put(3.9,3){\line(1,0){.2}}
\put(3.9,6){\line(1,0){.2}}
\put(3.9,7){\line(1,0){.2}}
\put(3.9,8){\line(1,0){.2}}
\put(3.5,4.8){$i$}
\end{picture}

\vspace*{3em}

\hspace*{9cm}\begin{picture}(8.5,8.5)
\put(-.5,4){\vector(1,0){9}}
\put(4,-.5){\vector(0,1){9}}
\put(0,3.9){\line(0,1){.2}}
\put(1,3.9){\line(0,1){.2}}
\put(2,3.9){\line(0,1){.2}}
\put(3,3.9){\line(0,1){.2}}
\put(5,3.9){\line(0,1){.2}}
\put(6,3.9){\line(0,1){.2}}
\put(7,3.9){\line(0,1){.2}}
\put(8,3.9){\line(0,1){.2}}
\put(5.1,3.5){$1$}
\put(3.9,5){\line(1,0){.2}}
\put(3.9,0){\line(1,0){.2}}
\put(3.9,1){\line(1,0){.2}}
\put(3.9,2){\line(1,0){.2}}
\put(3.9,3){\line(1,0){.2}}
\put(3.9,6){\line(1,0){.2}}
\put(3.9,7){\line(1,0){.2}}
\put(3.9,8){\line(1,0){.2}}
\put(3.5,4.8){$i$}
\end{picture}

\end{minipage}


\newpage
\mbox{}










\end{document}
