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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 1999}
\\
{\bf Final Examination}
\\
{\bf May 3, 1999}
\end{center}
\vspace*{.4cm}

\begin{tabular}{p{18cm}}\hline
\ \\
This Examination contains 12 problems on 12 sheets of paper 
including the front cover.
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}[t]{|c|c|c|}\hline
Question & Possible\quad &\ Actual\quad\rule[-.5cm]{0cm}{1.2cm} \\ \hline\hline
  1      &   15     &       \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      &   10     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
  6       &  10    &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
\end{tabular} \qquad
\begin{tabular}[t]{|c|c|c|}\hline
Question & Possible\quad &\ Actual\quad\rule[-.5cm]{0cm}{1.2cm} \\ \hline\hline
  7        &   15     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
  8        &   15     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
  9       &   15     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
  10      &   15     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
  11      &   10     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
  12        &   5     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
\end{tabular}
\\ \vspace*{.2ex}
\begin{tabular}[t]{|c|c|c|}\hline
\rule[-.5cm]{1cm}{0cm}\huge Total  \rule[-.5cm]{1cm}{0cm}
  & \huge\rule[-.5cm]{1cm}{0cm}150 \rule[-.5cm]{1cm}{0cm}& 
\rule[-.8cm]{0cm}{2cm}\rule{4.27cm}{0cm}
 \\ \hline
\end{tabular}
\end{center}

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
{\bf 1.} \begin{minipage}[t]{17.5cm} (15 points)
Let $\zeta = 1+i$. Compute the standard form $a+bi$ of the following
expressions. Simplify your answer.
\\[2ex]
(a) $2\zeta^2 +3\bar{\zeta}+4$
\vspace*{5cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{8cm}}}
\\[2ex]
(b)  ${\Ds {\Big(\frac{\zeta}{\sqrt2}\Big)^{100} }}$
\vspace*{5cm}

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

{\bf 2.} \begin{minipage}[t]{17.5cm}  (10 points)
Find all 6 roots of $z^{6} = -64$ and show them
in the picture.

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\put(3.9,8){\line(1,0){.2}}
\put(3.5,4.8){$i$}
\end{picture}

\end{minipage}
\newpage

{\bf 3.} \begin{minipage}[t]{17.5cm}  (15 points)
For each of the following problem decide whether $u(x,y)$ can be
the real part of an analytic function $f$. If so, find $v(x,y)$
so that $f=u+iv$ is analytic. Find also $f'$.
\\[2ex]
(a) $u(x,y) = x^3 - y^3$.
\vspace*{10cm}

(b) $u(x,y) = x^2 - y^2 $.


\end{minipage}
\newpage


%%%%%%%%%%%%%%%%%%%%%%%%%%
{\bf 4.} \begin{minipage}[t]{17.5cm}  (15 points)
Calculate all values of the following expressions and present the results in
standard form $a+bi$. Simplify your answer.
\\[2ex]
(a) ${\cos(2\pi i)}$
\vspace*{6.5cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{8cm}}}
\\[2ex]
(b) $(1-i)^{1+i}$
\vspace*{6.5cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{8cm}}}
\\[2ex]
(c) $\log(-3-i)$
\vspace*{6.5cm}

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

%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
{\bf 5.} \begin{minipage}[t]{17.5cm}  (10 points)
Find the radius of convergence of the following power series.
\\[2ex]

(a) ${\Ds \sum_{n=0}^\infty \frac{n^9(n+1)^8}{2^n} z^n}$ ,
\vspace*{10cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{8cm}}}
\\[2ex]
(b) ${\Ds \sum_{n=0}^\infty \frac{n}{(1+i)^n} (z-i)^n}$ ,
\vspace*{10cm}

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

%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
{\bf 6.} \begin{minipage}[t]{17.5cm}  (10 points)
(1) Find the Laurent series expansion about the point $z_0=0$ of the function
${\Ds f(z) =\frac{e^{2z}-2z-1}{z^3}}$. 

(2) Show the principal part explicitly. Find the region where the
Laurent series converges.

(3) Classify the type of singularity of this function at $z_0=0$
and find the residue.
\vspace*{10cm}

\end{minipage}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
{\bf 7.} \begin{minipage}[t]{17.5cm}  (15 points)
Compute the following integrals
\\[2ex]
(a) ${\Ds \int_{|z-3i|=3} \frac{e^z}{
(z^2+1)^2}dz}$.
\vspace*{10cm}
 
\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{8cm}}}
\\[2ex]
(b) ${\Ds \int_{|z|=100} \frac{z^2 e^{3z}}{z^2+3z+2}dz}$
\vspace*{10cm}


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

%%%%%%%%%%%%%%%%%%%%%%%
\newpage
{\bf 8.} \begin{minipage}[t]{17.5cm}  (15 points)
Compute the following integrals.
\\[2ex]
(a) ${\Ds \int_{-\infty}^\infty \frac{x^2}{1+x^4}dx}$.
\vspace*{10cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{8cm}}}
\\[2ex]
(b) ${\Ds \int_{-\infty}^\infty \frac{\sin x}{x^2+2x+2}dx}$
\vspace*{10cm}

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

%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
{\bf 9.} \begin{minipage}[t]{17.5cm}  

(a) (5 points). State PRECISELY Rouch\'e's Theorem.
\vspace*{5cm}

(b) (10 points).
Find the number of zeros of the polynomial 
$p(z)= z^6 +  z^3 + 8z^2  +1  $ in the annulus $\{1<|z|<2\}$.
\end{minipage}

%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
{\bf 10.} \begin{minipage}[t]{17.5cm}  

(a) (5 points). State PRECISELY the Riemann Mapping Theorem.
\vspace*{5cm}

(b) (10 points). Find a conformal mapping which maps the domain
$D=\{z \;\;|\;\; 0<\mbox{arg}z<3\pi/2\}$
(one-to-one and onto) to the unit disk.
\vspace*{10cm}

\end{minipage}

\newpage

{\bf 11.} \begin{minipage}[t]{17.5cm}

(10 points).  Determine whether you can use the Riemann Mapping Theorem
for the following domains, and JUSTIFY your answer.

(a) $D = \C\setminus\{yi\;\;|\;\; -\infty<y\le 1\}$.
\vspace*{10cm}

(b) $D = \{z\;\;|\;\; 0<|z|<1\}$.
\end{minipage}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
\newpage
 
{\bf 12.} \begin{minipage}[t]{17.5cm}
 
(5 points).  

Let $f(z)$ be a function defined on the closed unit disk
$\{z\;|\; |z|\le 1\}$.  Assume that 

(1).  $f(z)$ is analytic in the open
disk $\{z\;|\; |z|< 1\}$ and continuous on the closed disk
$\{z\;|\; |z|\le 1\}$,

(2).  $z_0=0$ is a zero of $f(z)$ of order $N$,  

(3). $|f(z)|\le 1$ on the circle $|z|=1$.

Prove that $|f(z)|\le |z|^N$ for all $z$ in the unit disc. 
\end{minipage}
\end{document}
