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\lhead{\bf Complex Analysis (MAA 4402 / 5404)}
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\rhead{Due Wednesday 1/29}
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%\noindent\emph{Answers should be submitted on a separate piece of paper. You need not attach this page, although if you do, then you will have the questions when your work is returned. Work which is sloppy or messy or that which is not written in a clear and coherent fashion will be marked down.}

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\begin{enumerate}

\Problem Let $S$ denote the sector in the complex plane defined by $r\le 1$ and $0\le\theta\le\pi/4$. Sketch the regions onto which $S$ is mapped by the following functions. \emph{No reasoning is required, just the sketches.}
	\begin{enumerate}
	\item[(a)] $f(z)=z^2$.
	\item[(b)] $f(z)=z^3$.
	\item[(c)] $f(z)=z^4$.
	\end{enumerate}

\end{enumerate}

\bigskip\bigskip\bigskip

\noindent There are many theorems which make computing limits easier. However, when a question asks you to compute a limit \emph{from the definition}, that means that to show that
\[
	\lim_{z\to z_0} f(z)=w_0,
\]
you must show that for every positive real number $\varepsilon>0$, there is another positive real number $\delta>0$ such that
\[
	|z-z_0|<\delta
	\quad\mbox{implies}\quad
	|f(z)-w_0|<\varepsilon.
\]

\bigskip\bigskip\bigskip

\begin{enumerate}

\Problem Using only the definition, show that $\ds\lim_{z\to z_0} \mbox{Re $z$}=\mbox{Re $z_0$}$.

\Problem Let $a$ and $b$ denote arbitrary complex constants. Using only the definition, show that $\ds\lim_{z\to z_0} (az+b)=az_0+b$.



\end{enumerate}

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