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\noindent
\parbox{1.95in}{\bf Math 5160}
\hfill {\Large \bf Homework \#8} \hfill
\parbox{1.95in}{\bf \hfill Do after Gamelin VI.2}

\vspace{0.2in}

%\noindent {\bf\large Name:} \underline{\qquad\sc\large This is my fabulous name! \qquad}
%
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\begin{description}
\item[\#1 Laurent Has Decomposed] Consider $f(z)=\dfrac{1}{z^2-z}$.
\begin{enumerate}[(a)]
\item Find \underline{all} of the Laurent expansions centered at $z=0$.
\item Find the Laurent expansion centered at $z=-1$ that converges at $z=1/2$.\\ \force\hfill Determine the largest open set where this expansion converges.
\end{enumerate}

\item[\#2 I am Just a Simple Pole in a Complex Plane] Find the isolated singularities of the following functions and determine whether they are removable, essential, or poles. For poles, determine the order and find the principal part at that pole.

\vspace{-0.15in}

$$ \mbox{(a)} \quad \dfrac{z}{(z^2-1)^2} \qquad \qquad
     \mbox{(b)} \quad \dfrac{ze^z}{z^2-1} $$
    
\item[\#3 Beware of Cancellation] Find the radius of convergence of the power series for the following functions,\\ \force\hfill expanded about the indicated point. 

$$ \mbox{(a)} \quad \dfrac{z-1}{z^4-1} \mbox{ about } z=3+i \qquad \qquad
     \mbox{(b)} \quad \dfrac{\cos(z)}{z^2-\pi^2/4} \mbox{ about } z=0 $$

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\item[\#UNUSED] Find the isolated singularities of the following functions and determine whether they are removable, essential, or poles. For poles, determine the order and find the principal part at that pole. 

$$ \mbox{(a)} \quad \dfrac{z}{(z^2-1)^2} \qquad \qquad
     \mbox{(b)} \quad \dfrac{ze^z}{z^2-1} \qquad \qquad
     \mbox{(c)} \quad \dfrac{\cos(z)}{z^2-\pi^2/4} \qquad \qquad
     \mbox{(d)} \quad e^{1/(z^2+1)} $$
    
{}[This is Gamelin VI.2 \#1a,b,f,i]


