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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 2000}
\\
{\bf Exam 2}
\\
{\bf March 27, 2000}
\end{center}
\vspace*{.4cm}

\begin{tabular}{p{18cm}}\hline
This
\rule[0cm]{0cm}{.8cm}
Examination contains 7 problems. 
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      &   10     &       \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      &   15     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
  7      &   25     &       \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} (10 points)
Without evaluating, find the radius of convergence of the 
power series (centered at $z_0$) of the following functions.
\\[2ex]
(a) $\Ds f(z)= \frac{e^z}{z^2-4}, \;\; z_0=0.$

\vspace*{6.5cm}

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

(b)
$\Ds g(z) = \frac{e^z}{z^2-4}, \;\; z_0= 3. $

\vspace*{6.5cm}

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

(c)
$\Ds h(z) = \frac{e^z}{z^2-4}, \;\; z_0= i. $

\vspace*{6.5cm}

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

\end{minipage}

\newpage
{\bf 2.} \begin{minipage}[t]{17.5cm} (10 points)

(a) State PRECISELY Liouville's theorem.

\vspace*{5.5cm}

(b) Suppose that $f$ is an entire function and $Re(f)\le C$ for
all $z$. Prove that $f$ is a constant.

\end{minipage}

\newpage
{\bf 3.} \begin{minipage}[t]{17.5cm} (15 points)
Find the {\bf radius of convergence} of the following power series.
\\[2ex]
(a) ${\Ds \sum_{n=0}^{\infty} (n+1)(n+3)(z-2-i)^n}$

\vspace*{6cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{8cm}}}
\\[2ex]
(b) ${\Ds \sum_{n=0}^{\infty} \frac{i^n n^2}{5^{n}} (z+2)^{n}}$

\vspace*{6cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{8cm}}}
\\[2ex]
(c) ${\Ds \sum_{n=0}^{\infty} \frac{4n}{3^n}z^{2n}}$

\vspace*{6cm}

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

\newpage
{\bf 4.} \begin{minipage}[t]{17.5cm} (15 points)

Find the power series about the origin for the given
function.

(a) $\Ds \frac{2+z}{2-z} $

\vspace*{10cm}
 
(b) $\Ds  z^4e^{-2z} $



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

\newpage
{\bf 5.} \begin{minipage}[t]{17.5cm} (10 points)

True or False questions. 

Determine whether the following statement is true or false.

(a) If a function $f(z)$ has a continuous first order
 derivative on a domain $D$, then derivatives of $f(z)$ of
 any orders exist.

\vspace*{3.5cm}

(b) If a function $f(z)$ has a power series expansion
 in the disc $\{|z-z_0|<R\}$, then $f(z)$
 is an analytic function in this disc $\{|z-z_0|<R\}$.

\vspace*{3.5cm}

(c) For an analytic function $f$ on a domain, if $Re(f)$ is
a constant, then $Im(f)$ is also a constant.

\vspace*{3.5cm}

(d) For any analytic function $f$ on a domain $D$, and a 
smooth (but not closed) curve inside $D$
$\gamma: z=\gamma(t), a\le t\le b, \;\;
\gamma(a)\neq\gamma(b)$, we have
$$
 \int_\gamma f(z) dz = 0.
$$

\vspace*{3.0cm}

(e) If $f(z)$ is an analytic function on a domain, then its conjugate
$\overline{f(z)}$ is also an analytic function.
 
\end{minipage}

\newpage
{\bf 6.} \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. 
\\[2ex]
(a) $u(x,y)= 3x^2 y^2$

\vspace*{9cm}

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

\vspace*{9cm}

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


{\bf 7.} \begin{minipage}[t]{17.5cm} (25 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 you used to obtain your answer.
\\[2ex]
(a) ${\Ds \int_{|z|=4}\frac{1}{z^2+4}dz}$

\vspace*{10cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{8cm}}}
\\[2ex]
(b) ${\Ds \int_{|z|=5} \frac{ e^{3z} }{(z-3)(z+7)^2}dz}$

\vspace*{10cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{8cm}}}
\end{minipage}
\newpage
\begin{minipage}[t]{17.5cm}{}
(c) ${\Ds \int_{|z-i|=20} \frac{e^{2z}}{z^2+3z+2}dz}$

\vspace*{10cm}


\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{8cm}}}
\\[2ex]
(d) ${\Ds \int_0^{2\pi} \frac{d\theta}{2+\cos\theta} }$
\vspace*{10cm}


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