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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 Exam 3}
\\
{\bf April 12, 1999}
\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.
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
Total    & 100\mbox{\  \ } & \rule[-.5cm]{0cm}{1.2cm} \\ \hline
\end{tabular}
\end{center}

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

\newpage
{\bf 1.} \begin{minipage}[t]{17.5cm} (20 points)
For each of the following problem, (1) Find all singularities of the
following functions in the given domain, (2) Determine the type (removable
singularity, pole, essential singularity) of each of the singularities,
(3) the order of each of the poles, and (4) the residue at poles.
\\[2ex]
(a) $\Ds f(z)=\frac{1}{(z^2+4z+5)^2}, \;\;z\in \C$.

\vspace*{9cm}

(b) ${\Ds f(z)=\frac{z^7}{1+z^{9}} }, \;\;\;\; $ Re$\{z\}>0$,  Im$\{z\}>0$.

\vspace*{9cm}

\end{minipage}

\newpage
{\bf 2.} \begin{minipage}[t]{17.5cm} (20 points)
Find (1) the Laurent-series, and  (2) the residue of the following
function at $z_0$.

(a) ${\Ds f(z) = \frac{z^2}{z^2-1}}$ at $z_0=1$.
\\[2ex]
\vspace*{9.5cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{10cm}}}
\\[2ex]
(b) ${\Ds f(z)=e^{2z+3}+ z e^{1/(z-2)} } $ at $z_0=2$.
\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) ${\Ds \int_{-\infty}^{\infty}\frac{1}{x^4+1}dx}$
\vspace*{9.5cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{8cm}}}
\\[2ex]
(b) ${\Ds \int_{-\infty}^{\infty} \frac{\cos 2x}{x^2+1}dx}$

\vspace*{9.5cm}

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

\newpage
{\bf 4.} \begin{minipage}[t]{17.5cm} (20 points)
(1). Draw a correct path in the complex plane for each of the following problem
that will evaluate the real integrals. (2). Find the function $f(z)$
that you will integrate on the path.
\\[2ex]

(a) ${\Ds \int_{-\infty}^\infty \frac{x^3\sin x }{x^6 + 3x^4+ 3x^2+1}dx}$

\vspace*{5cm}

(b) ${\Ds \int_{0}^\infty \frac{\sqrt{x} }{x^2+4x + 9}dx}$

\vspace*{5cm}
 
(c) ${\Ds \int_{0}^\infty \frac{1-\cos 4x}{x^2}dx}$

\vspace*{5cm}

(d) ${\Ds \int_{0}^\infty \frac{\log x}{ x^2 + 2x + 10}dx}$

\vspace*{5cm}
\end{minipage}

\newpage
{\bf 5.} \begin{minipage}[t]{17.5cm} 

(a). (6 points). State PRECISELY the Liouville's theorem. (Assumptions
and conclusions).
\vspace*{5cm}

(b). (6 points). Suppose that $f(z)$ is an analytic function
on the entire complex plane such that Re$\{f(z)\}\le c$ for all
$z\in \C$, prove that $f(z)$ is constant.
\vspace*{9cm}

(c). (4 points).  Suppose that $f(z)$ is an analytic function
on the entire complex plane such that $|f(z)|\ge 1$ or all
$z\in \C$, prove that $f(z)$ is constant.
\end{minipage}

\newpage
 
(d). (4 points). Suppose that $f(z)$ is an analytic function
on the entire complex plane such that the range of $f$ is not
dense in $\C$, use (c) to prove that $f(z)$ is constant.


\end{document}
