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\noindent
\parbox{2.25in}{\bf Math 4010/5160}
\hfill {\Large \bf Homework \#7} \hfill
\parbox{2.25in}{\bf \hfill Due: Tues., Apr. $17^\mathrm{th}$, 2018}

\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 Bumping Up Against Infinity] Find the {\bf radius of convergence} of the power series for the following functions if they were expanded about the given point. {}[This is Gamelin V.4 \#1a,b,c]

\vspace{-0.15in}

$$ \mbox{(a)} \quad \dfrac{1}{z-1} \mbox{ about } z=i \qquad \qquad
     \mbox{(b)} \quad \sec(z) = \dfrac{1}{\cos(z)} \mbox{ about } z=0 \qquad \qquad
     \mbox{(c)} \quad \mathrm{sech}(z) = \dfrac{1}{\cosh(z)} \mbox{ about } z=0 $$

\item[\#2 Standard Series Manipulation] Find the power series expansion of the principal branch of $\mathrm{arctan}(z)$ expanded about $z=0$. What is the radius of convergence of this series? {\it Hint:} Find an expansion for the derivative of arctan using a geometric series and then integrate it.
{}[This is Gamelin V.4 \#7]

\item[\#3 Infinite Power] Find the power series expansion about $z=\infty$ for $\dfrac{z^2}{z^3-1}$. Where does this series converge?\\
{}[This is part of Gamelin V.5 \#1]

\item[\#4 Who Needs Polynomial Long Division?] Calculate up to (and including) the fifth order terms in the power series expansion about $z=0$ (where do these expansions converge?): {}[This is parts of Gamelin V.6 \#2 and \#3]

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$$ \mbox{(a)} \quad \dfrac{z}{\sin(z)} \qquad \qquad \mbox{(b)} \quad \dfrac{e^z}{1+z}$$

\item[\#5 Establishing Order] For each of the following functions, first, find the zeros and their orders. Next, determine if they are analytic at $\infty$ and determine orders of any zeros there.
{}[This is Gamelin V.7 \#1d,f and \#2d,f]

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$$ \mbox{(a)} \quad \cos(z)-1 \qquad \qquad
     \mbox{(b)} \quad \dfrac{\cos(z)-1}{z} $$

\item[\#6 This is Easy -- Don't Make it Hard] Show that all of the zeros of $\sin(z)$ and $\tan(z)$ are simple.\\
{}[This is Gamelin V.7 \#3]

\item[\#7 Laurent Has Decomposed] Consider $f(z)=\dfrac{1}{z^2-z}$ {}[This is Gamelin VI.1 \#1a and \#2a]
\begin{enumerate}[(a)]
\item Find all of the Laurent expansions centered at $z=0$.
\item Find the Laurent expansion centered at $z=-1$ that converges at $z=1/2$. Determine the largest open set where this expansion converges.
\end{enumerate}

\item[\#8 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. {}[This is Gamelin VI.2 \#1a,b]

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$$ \mbox{(a)} \quad \dfrac{z}{(z^2-1)^2} \qquad \qquad
     \mbox{(b)} \quad \dfrac{ze^z}{z^2-1} $$
    
\item[\#9 Beware of Cancellation] Find the radius of convergence of the power series for the following functions, expanded about the indicated point. {}[This is Gamelin VI.2 \#2a,b]

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

\end{description}

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\item[\#1] Find the domain of convergence for the given power series.

$$ \mbox{(a)}\quad \sum\limits_{m=0}^\infty 2^m(z-2)^m \qquad \qquad
     \mbox{(b)} \quad \sum\limits_{m=1}^\infty \dfrac{(z+i)^m}{m^2} \qquad \qquad
     \mbox{(c)} \quad \sum\limits_{n=1}^\infty n^n(z-3)^n $$

{}[This is Gamelin V.3 \#2c,d,e]

\item[\#3] Find the power series expansions about $z=\infty$ for the following functions (where do these expansions converge?):

$$ \mbox{(a)} \quad \dfrac{1}{z^2+1} \qquad \qquad
     \mbox{(b)} \quad \dfrac{z^2}{z^3-1} \qquad \qquad
     \mbox{(c)} \quad e^{1/z^2}$$

{}[This is Gamelin V.5 \#1]

\item[\#8] 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]
