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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 2}
\\
{\bf October 17, 1997}
\end{center}
\vspace*{.4cm}

\begin{tabular}{p{18cm}}\hline
This
\rule[0cm]{0cm}{.8cm}
Examination contains 6 problems on 8 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      &   10     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
  3      &   20     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
  4      &   15     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
  5      &   15     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
  6      &   20     &       \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)
Compute the following line integrals.
\\[2ex]
(a) ${\Ds \int_{\gamma}|z|(z+1)dz},$ where $\gamma$ is the straight line
segment joining
$-2i$ to $2i$.
\vspace*{9.5cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{8cm}}}
\\[2ex]
(b) ${\Ds \int_{\gamma}\im(z)dz},$ where $\gamma$ is the semicircle joining
$-2i$ to $2i$ and passing through $2$.

\vspace*{9.5cm}

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

\newpage
{\bf 2.} \begin{minipage}[t]{17.5cm} (10 points)
Let $\gamma$ be the circle with radius $R$ and center $3-4i$  oriented
counterclockwise.
\\
Compute the following line integrals
\\[2ex]
(a) ${\Ds \int_{\gamma}\frac{2iz}{z-3+4i}dz}$

\vspace*{9.5cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{8cm}}}
\\[2ex]
(b) ${\Ds \int_{\gamma}\frac{2i}{(z-3+4i)^2}dz}$
\vspace*{9.5cm}

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

\newpage
{\bf 3.} \begin{minipage}[t]{17.5cm} (20 points)
Find the radius of convergence of the following power series.
\\[2ex]
(a) ${\Ds \sum_{n=0}^{\infty} 5^{n^2}(z-1)^n}$

\vspace*{4cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{8cm}}}
\\[2ex]
(b) ${\Ds \sum_{n=0}^{\infty} 3^n(z+6i)^n}$

\vspace*{4cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{8cm}}}
\\[2ex]
(c) ${\Ds \sum_{n=0}^{\infty} z^{n^2}}$

\vspace*{4cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{8cm}}}
\\[2ex]
(d) ${\Ds \sum_{n=0}^{\infty} \frac{z^n}{n^n}}$

\vspace*{4cm}

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

\newpage
{\bf 4.} \begin{minipage}[t]{17.5cm} (15 points)
Knowing that ${\Ds \cos z=\sum_{n=0}^{\infty}\frac{(-1)^n z^{2n}}{(2n)!}}$,
find
the first four terms of the power series expansion of the function
${\Ds \frac{\cos z}{1-z}}$.
Where does the latter expansion converge?

\vspace*{17cm}

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

\newpage
{\bf 5.} \begin{minipage}[t]{17.5cm} (15 points)
For each of the following functions $u(x,y)$ decide whether it 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 the derivative $f'$.
\\[2ex]
(a) $u(x,y)=e^{-x}\cos y$

\vspace*{9cm}

%\mbox{}\hfill
{\bf Answer:} \underline{\mbox{\hspace*{16cm}}}
\\[3ex]
(b) ${\Ds u(x,y)=x^2y+\frac{1}{3}y^3}$

\vspace*{9cm}

%\mbox{}\hfill
{\bf Answer:} \underline{\mbox{\hspace*{16cm}}}
\end{minipage}


{\bf 6.} \begin{minipage}[t]{17.5cm} (20 points)
Let $|z-a|=r$ denote the circle with radius $r$ and center $a$ oriented
counterclockwise.
\\
Find the values of the following integrals.
Indicate which theorem or device you used to obtain your answer.
\\[2ex]
(a) ${\Ds \int_{|z-\frac{1}{2}|=1}\frac{1}{z+2i}dz}$

\vspace*{4cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{8cm}}}
\\[2ex]
(b) ${\Ds \int_{|z|=2} \frac{4i-5}{(z-\frac{1}{2})^{15}}dz}$

\vspace*{4cm}

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

\vspace*{4cm}

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

\vspace*{4cm}


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