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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 Final Examination}
\\
{\bf December 17, 1997}
\end{center}
\vspace*{.4cm}

\begin{tabular}{p{18cm}}\hline
This
\rule[0cm]{0cm}{.8cm}
Examination contains 14 problems on 11 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}[t]{|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      &   10     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
  4      &   10     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
  5      &   15     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
  6      &   15     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
  7       &  \mbox{\ } 5    &       \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
  8        &   10     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
  9        &   10     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
 10       &   10     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
 11       &   10     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
  12      &   15     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
  13       &   10     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
  14        &   10     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline\hline
Total    & 150\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)
Let $\zeta = 3-4i$. Compute the standard form $a+bi$ of the following
expressions.
\\[2ex]
(a) $\zeta^2 +4\bar{\zeta}+2$
\vspace*{5cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{8cm}}}
\\[2ex]
(b)  ${\Ds \frac{\zeta}{1+2i}}$
\vspace*{5cm}

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

{\bf 2.} \begin{minipage}[t]{17.5cm}  (10 points)
Let $u=x^3-kx^2-3xy^2+x+4y^2$.
Find $k$ so that $u$ is the real part of an analytic function $f$.
Determine also $f$ and $f'$.
\vspace*{9cm}

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

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
{\bf 3.} \begin{minipage}[t]{17.5cm}  (10 points)
Define all possible analytic logarithms on the domain
$D=\C\backslash\{t(-1-i)\,|\,t\geq0\}$.
Give the values at the point $2i$ for all these logarithms.
\vspace*{7cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{8cm}}}
\end{minipage}
\\[2ex]
{\bf 4.} \begin{minipage}[t]{17.5cm}  (10 points)
Find all 8th roots $z_0,\,z_1,\ldots,\,z_7$ of $-16$ and show each of  them
in the
picture.

\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}

\end{minipage}

%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
{\bf 5.} \begin{minipage}[t]{17.5cm}  (15 points)
Calculate all values of the following expressions and present the results in
standard form $a+bi$.
\\[2ex]
(a) ${\Ds \Big(\frac{1}{\sqrt{2}}(-1+i)\Big)^i}$
\vspace*{6.5cm}

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

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{8cm}}}
\\[2ex]
(c) $\log(-4)$
\vspace*{6.5cm}

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

%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
{\bf 6.} \begin{minipage}[t]{17.5cm}  (15 points)
Compute the following integrals.
\\[2ex]
(a) ${\Ds \int_{\gamma}2z\im(z)dz}$ ,
where $\gamma$ is the straight line segment joining $1$ to $3i$.
\vspace*{8cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{8cm}}}
\\[2ex]
(b) ${\Ds \int_{\gamma} \overline{z}^2 dz}$, where $\gamma$ is the
semicircle from $3$ to
$-3$ through $-3i$
\vspace*{8cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{8cm}}}
\end{minipage}
\\[2ex]
{\bf 7.} \begin{minipage}[t]{17.5cm}  (5 points. No partial credit!)
Which of the following is true? Circle your answer!
\\[2ex]
(a) $\overline{e^{iz}}=-e^{iz}$\qquad
(b) $\overline{e^{iz}}= e^{i\overline{z}}$\qquad
(c) $\overline{e^{iz}}= e^{-i\overline{z}}$\qquad
(d) $\overline{e^{iz}}= e^{-iz}$\qquad
(e) $\overline{e^{iz}}=-e^{-iz}$
\end{minipage}

%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
{\bf 8.} \begin{minipage}[t]{17.5cm}  (10 points)
Find theLaurent series expansion about the point $z_0=0$ of the function
${\Ds f=\frac{e^z-z-1}{z^4}}$. Show the principal part explicitly.
\vspace*{10cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{8cm}}}
\end{minipage}
\\[2ex]
{\bf 9.} \begin{minipage}[t]{17.5cm}  (10 points)
Let $f=u+iv$ be an entire function with $u=\re(f)$ and $v=\im(f)$.
Suppose there exists a constant $c\in\R$ so that $u(z)\leq c$ for all $z\in\C$.
Consider the function $g(z)=e^{f(z)}$.
\\[2ex]
(a)  Calculate $|g(z)|$.
\vspace*{2.5cm}

(b)  What follows from the above assumption about $u$ for $|g(z)|$?
\vspace*{2.5cm}

(c) What can be concluded for the function $g$? Indicate which theorem you
are using.
\vspace*{2.5cm}

(d) What can be said about the function $f$?
\end{minipage}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
{\bf 10.} \begin{minipage}[t]{17.5cm}  (10 points)
Compute the  integral ${\Ds
\int_{-\infty}^{+\infty}\frac{x^2-2}{x^4+8x^2+15}dx}$.
\vspace*{10.6cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{8cm}}}
\end{minipage}
\\[2ex]
{\bf 11.} \begin{minipage}[t]{17.5cm}  (10 points)
Compute the integral ${\Ds \int_{0}^{2\pi} \frac{d\theta}{5+3\cos\theta}}$.
\vspace*{10.6cm}

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

%%%%%%%%%%%%%%%%%%%%%%%
\newpage
{\bf 12.} \begin{minipage}[t]{17.5cm}  (15 points)
Compute the following integrals.
\\[2ex]
(a) ${\Ds \int_{\gamma}\frac{\sin z}{(z-3)^2(z+2i)}dz}$, where $\gamma$ is the
polygonal curve connecting the points
\\[1ex]
$z_0=1,\,z_1=2+2i,\,
z_3=i,\,z_4=-2+2i,\,z_5=-1,\,z_6=-2-2i,\,z_7=-i,\,z_8=2-2i,\,z_9=z_0$
\\[1ex]
(in that order).  Show $\gamma$ in the diagram.
%rectangle with vertices $1+i,\,-1+i,\,-1-i$, and $1-i$.


\unitlength.5cm
\mbox{}\hfill\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(5.1,3.4){\small{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){\it\small i}
\end{picture}
\vspace*{1cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{8cm}}}
\\[2ex]
(b) ${\Ds \int_{|z|=2}\frac{z}{z^4-1}dz}$
\vspace*{6.5cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{8cm}}}
\\[2ex]
(c) ${\Ds \int_{|z-2|=2}\frac{5z^2-3z+2}{(z-1)^3}dz}$
\vspace*{6cm}

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

%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
{\bf 13.} \begin{minipage}[t]{17.5cm}  (10 points)
Find all singularities of the given function $f$.
Compute the value of $f$ at each of the removable singularities and the
residue of $f$
at each of the poles.
\[
   f=\frac{e^{2\pi i z}-1}{z(z-1)(4z^2-1)}
\]
\end{minipage}

%%%%%%%%%%%%%%%%%%%%%%%%%
\newpage
{\bf 14.} \begin{minipage}[t]{17.5cm}  (10 points)
\\[2ex]
(a) Compute the number of zeros of $f(z)=z^5-10z^2+4iz-3+2i$ in the annulus
$1<|z|<3$.
\vspace*{10cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{8cm}}}
\\[2ex]
(b) Find the number of solutions of the equation $z^6-1=z^5+z^4-5iz+1$ in
the punctured
disk
\\
\phantom{(b)} $0<|z|<2$.
\vspace*{10cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{8cm}}}
\end{minipage}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\wegdamit{
{\bf 12.} \begin{minipage}[t]{17.5cm} (10 points; no partial credit!)
Suppose $f$ is an analytic and bounded function on the punctured disk
$\{z\in\C\,|\,0<|z|<5\}$.
What can be concluded about the point $z_0=0$? Circle your answer.
\\[2ex]
\begin{tabular}{lll}
A. $z_0$ is an essential singularity.     & B.  $z_0$ is removable. $\qquad$
C.  $z_0$ is a singularity of any type.  \\[1.5ex]
D.  $z_0$ is a pole. &
E. $z_0$ is removable and the value of $f$ at $z_0$ is zero.
\end{tabular}
\end{minipage}
}
\end{document}
