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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 2}
\\
{\bf March 3, 1999}
\end{center}
\vspace*{.4cm}

\begin{tabular}{p{18cm}}\hline
This
\rule[0cm]{0cm}{.8cm}
Examination contains 8 problems on 9 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      &   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      &   15     &       \rule[-.5cm]{0cm}{1.2cm} \\ \hline
  8      &   10     &       \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)
Compute the following line integrals.
\\[2ex]

(a) ${\Ds \int_{\gamma} \Big(\frac1z + z\Big) dz},$
where $\gamma$ is the semicircle joining
$5i$ to $-5i$ and passing through $-5$.

\vspace*{9.5cm}

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

(b)
${\Ds \int_{\gamma} \Big(\frac1{z -1-i} \Big)^4 dz}$, where
$\gamma$ is the semicircle with radius $5$ and center $1+i$ 
joining  $6+i$ and $-4+i$ and passing through $1+6i$.

\vspace*{9.5cm}

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

\end{minipage}

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

(a) Assume that $|u(z)|\le C$ for all $z$. For large $R$, estimate the integral
$${\Ds \int_{\gamma_R} \frac{(z-4i)u(z)}{(z+3)^{3} } dz},$$
where $\gamma_R$ is the circle $|z|=R$.

\vspace*{12.5cm}

(b) Find the limit
$$\lim_{R\to\infty} \int_{\gamma_R} \frac{(z-4i)u(z)}{(z+3)^{3} } dz
 . $$

\end{minipage}

\newpage
{\bf 3.} \begin{minipage}[t]{17.5cm} (15 points)
Find the {\bf radius of convergence} and {\bf the center $z_0$}
 of the following power series.
\\[2ex]
(a) ${\Ds \sum_{n=0}^{\infty} (2i)^{n}(z+i)^n}$

\vspace*{6cm}

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

\vspace*{6cm}

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

\vspace*{6cm}

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

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

Find a ``closed form'' (that is, a simple expression) for
each of the given series.

(a) $\Ds \sum_{n=2}^\infty n(n-1) z^n $

\vspace*{8cm}
 
(b) $\Ds \sum_{n=1}^\infty n^2 z^n $



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

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

 Find the values of the integrals. Indicating which theorem
you used to obtain your answer.

(a) $$\Ds \int_{\gamma} e^{i z} dz,$$ where $\gamma$ is any piecewise smooth
 curve joining $0$ to $4+\pi i$.

\vspace*{8.3cm}

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

\vspace*{3ex}
(b) 
 $$ \int_\gamma \frac1 z dz,$$
where $\gamma$ is any piecewise smooth curve which lies in the upper half
plane $\{{\rm Im}\; z>0\}$ joining $1+i$ to $3i$.

\vspace*{8.3cm}

%\mbox{}\hfill
{\bf Answer:} \underline{\mbox{\hspace*{16cm}}}
\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. Find also the derivative $f'$.
\\[2ex]
(a) $u(x,y)=x^4-y^4$

\vspace*{9cm}

%\mbox{}\hfill
{\bf Answer:} \underline{\mbox{\hspace*{16cm}}}
\\[3ex]
(b) ${\Ds u(x,y)= 6xy}$

\vspace*{9cm}

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


{\bf 7.} \begin{minipage}[t]{17.5cm} (15 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) ${\Ds \int_{|z|=5}\frac{1}{z-4i}dz}$

\vspace*{6cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{8cm}}}
\\[2ex]
(b) ${\Ds \int_{|z|=4} \frac{ e^z \cos (\pi z) }{(z-2)(z+6)^2}dz}$

\vspace*{6cm}

\mbox{}\hfill{\bf Answer:} \underline{\mbox{\hspace*{8cm}}}
\\[2ex]
(c) ${\Ds \int_{|z-i|=100} \frac{3z^2+ 2z + 1}{(z-i)(z+i)}dz}$

\vspace*{6cm}


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

\newpage
{\bf 8.} \begin{minipage}[t]{17.5cm} (10 points)
Suppose that $g$ is twice differentiable with continuous 2nd order
derivatives. $\Ds P=\frac{\p g}{\p x}$ and $\Ds Q=\frac{\p g}{\p y}$
on an open disc $D$.  Prove that, for any piecewise smooth, closed
simple curve $\gamma$ inside $D$,
$$
 \int_{\gamma} \Big( P dx + Q dy\Big) = 0.
$$
\end{minipage}
\end{document}
