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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 3}
\\
{\bf November 21, 1997}
\end{center}
\vspace*{.4cm}

\begin{tabular}{p{18cm}}\hline
This
\rule[0cm]{0cm}{.8cm}
Examination contains 5 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      &   20     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
  2      &   20     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
  3      &   20     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
  4      &   20     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
  5      &   20     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline\hline
%  6      &   10     &       \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} (20 points)
Find all singularities of the following functions.
Determine the type of each of the singularities and the order of each of
the poles.
Find the value of the function at each of the removable singularities and the
residue at each of the poles.
\\[2ex]
(a) $f(z)=e^{z^2+z+3}$

\vspace*{8cm}

\wegdamit{
(b) ${\Ds f(z)=\frac{\cos\pi z-1}{z^2(z-1)}}$

\vspace*{9.5cm}
}

(b) ${\Ds f(z)=\frac{z-2}{\sin \pi z}}$
\end{minipage}

\newpage
{\bf 2.} \begin{minipage}[t]{17.5cm} (20 points)
Find the principal part of the Laurent-series and the residue of the following
functions at the given point.
\\[2ex]
(a) ${\Ds f(z)=\frac{e^{iz}}{(z^2+4)^2}}\ $ at $z_0=2i$.

\vspace*{9.5cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{10cm}}}
\\[2ex]
(b) ${\Ds f(z)=\frac{4z^2-10z+1}{(z-1)^2}}\ $ at $z_0=1$.
\vspace*{9.5cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{10cm}}}
\end{minipage}

\newpage
{\bf 3.} \begin{minipage}[t]{17.5cm} (20 points)
Compute the following real integrals.
\\[2ex]
(a) ${\Ds \int_{-\infty}^{\infty}\frac{x \sin x}{x^2+9}dx}$
\vspace*{9.5cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{8cm}}}
\\[2ex]
(b) ${\Ds \int_{-\infty}^{\infty} \frac{x^2}{x^4+5x^2+6}dx}$

\vspace*{9.5cm}

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

\newpage
{\bf 4.} \begin{minipage}[t]{17.5cm} (20 points)
Find the values of the following integrals and present them in standard
form $a+ib$.
\\[2ex]
(a) ${\Ds \int_{|z|=1}\frac{e^z+2z}{z^5}dz}$

\vspace*{10cm}
\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{8cm}}}
\\[2ex]
%(b) ${\Ds \int_{|z-\pi|=1}\frac{\cos z}{(z-\pi)^6}dz}$
%
%\vspace*{7cm}
%\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{8cm}}}
%\\[2ex]
%(b)  ${\Ds \int_{|z-\frac{1}{2}|=1}\frac{z+1}{(z^2-1)^2}dz}$
%
%\vspace*{6.5cm}
%\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{8cm}}}
%\\[2ex]
(b) ${\Ds \int_{|z-1-i|=\frac{3}{2}}\frac{\cos \pi z}{z^4-1}dz}$

\vspace*{10cm}
\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{10cm}}}
\end{minipage}

\newpage
{\bf 5.} \begin{minipage}[t]{17.5cm} (20 points)
Find the zeros of the  function
\[
  f(z)=(\cos z -1)^2\sin z
\]
 and determine the  order of each zero.

%\vspace*{12cm}
%\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{10cm}}}
\end{minipage}
\wegdamit{
\\[3ex]
{\bf 6.} \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}
