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\begin{document} 
{\bf Name:} \underline{\mbox{\hspace*{7cm}}} 
\hfill{\bf Instructor:} Bei Hu 
 
\begin{center} 
{\bf Mathematics 425: Complex Variables } 
\\ 
{\bf Spring Semester 2000} 
\\ 
{\bf Exam 1} 
\\ 
{\bf February 18, 2000} 
\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 $z=1+2i$, $w=2+i$, 
compute 
$\bar{z}w+2z\bar{w}$ and put your answer in the form of $a+bi$. 
\vspace*{5.8cm} 
 
\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{8cm}}} 
\\[2ex] 
(b) Compute ${\Ds \Big(\frac{1-i}{\sqrt2}\Big)^{80}}$. \vspace*{5.8cm} 
 
\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{8cm}}} 
\end{minipage} 
\\[3.5ex] 
{\bf 2.} \begin{minipage}[t]{17.5cm} (10 points) Let $\alpha$ be a complex 
number such that $|\alpha|<1$. Show that the set of all 
$z$ with $|z-\alpha|=|1-\bar{\alpha}z|$  is the circle 
$\{z:|z|=1\}$. 
\vspace*{7cm} 
 
\end{minipage} 
 
\newpage 
{\bf 3.} \begin{minipage}[t]{17.5cm} (15 points) Find all 3 roots of $z^3= 
-5-5i$. \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) $\Ds f(z)= \frac{z^3+i}{z-i}$, $z_0=i$. Find 
$\Ds \lim_{z\to z_0} f(z)$. 
\vspace*{5.8cm} 
 
\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{8cm}}} 
\\[2ex] 
(b) 
 Find $\Ds \lim_{n\to\infty}  \Big(\frac{4+3i}{5}\Big)^n$. 
\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 convex?} Is the {\bf interior} of the set {\bf 
connected?} Show the set and sketch your ideas for the answers on the diagram. 
\\[2ex] 
(a) $\{z=x+iy,|\, -4<x<4, -2<y< x/4 \}$. 
 
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\put(3.5,4.8){$i$} 
\end{picture} 
\\[6ex] 
{\bf Answer:} \underline{\mbox{\hspace*{8cm}}} 
\\[6ex] 
(b) $\{z=r e^{i\theta}\,|\, 1< r < 4, -\pi/4\le\theta\le 3\pi/4\}$. 
 
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\\[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) $(1+i)^{1+i}$ \vspace*{4.0cm} 
 
\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{8cm}}} 
\\[2ex] 
(b) $\log(-1)$ \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) $\Ds \sum_{n=1}^\infty \frac{n \; i^n}{2^n}$ \vspace*{4.0cm} 
 
\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{8cm}}} 
\\[2ex] 
(b) $\Ds \sum_{n=1}^\infty \frac{n!(\sqrt3+i)^n}{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) 
 
Compute the integral 
$$\int_\gamma \frac1{z^2} dz, $$ 
where $\gamma$ is the semicircle from $2i$ to $-2i$, passing through $-2$. 
\end{minipage} 
 
 
\newpage 
\mbox{} 
 
 
 
 
 
 
 
 
 
 
\end{document} 
