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\noindent
\parbox{1.95in}{\bf Math 5160}
\hfill {\Large \bf Homework \#6} \hfill
\parbox{1.95in}{\bf \hfill Do after Gamelin V.1}

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%\noindent {\bf\large Name:} \underline{\qquad\sc\large This is my fabulous name! \qquad}
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\begin{description}
\item[\#1 Liouville Slugger] Let $u$ be a (real valued) harmonic function on the entire complex plane. Assume that $u$ is bounded above. Show that $u$ is constant.

{\it Note:} We discussed harmonic functions and a maximum modulus result earlier. However, I would like you to approach the problem as follows:
(1) Express $u$ as part of an analytic function (Why/how can this be done?). (2) Next, exponentiate and use Liouville's Theorem. (3) Make sure you justify your use of any theorems (i.e., Are the hypotheses satisfied?). 

\item[\#2 Lines Don't Matter] Let $L$  be a line in the complex plane. Suppose that $f(z)$ is continuous on a domain $D$ and analytic on $D-L = \{ z \in D \;|\; z \not\in L \}$.  Show that $f(z)$ is analytic on $D$. 

{\it Hint/Suggestion:} Use (don't reprove) the theorem in Gamelin about the special case of $L=\mathbb{R}$.

\item[\#3 More--rara Fun] State a version of Morera's theorem which uses triangles instead of rectangles. Then use Gamelin's version of Morera's theorem (page 119) to prove your version. 

{\it Note:} Make your theorem strong like Gamelin's. He doesn't require verification around all rectangles -- just certain kinds. What nice kind of triangles can you restrict to?

\item[\#4 Fancy Notation] Show that $\displaystyle \dfrac{\partial^2}{\partial x^2}+\dfrac{\partial^2}{\partial y^2} = 4\dfrac{\partial^2}{\partial z \partial \bar{z}}$. 

{\it Note:} You may (and should) assume that mixed second partials are continuous to prove the above identity.

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Next, let $h$ be a smooth complex-valued function defined on some complex domain $D$.

\begin{enumerate}[(a)]
\item Show that $h$ is harmonic on $D$ if and only if $\displaystyle \dfrac{\partial^2 h}{\partial z \partial \bar{z}} = 0$ on $D$.
\item Show that $h$ is harmonic on $D$ if and only if $\displaystyle \dfrac{\partial h}{\partial z}$ is analytic on $D$.
\end{enumerate}

\item[\#5 Some Analysis] Let $z_k$ be a sequence of complex numbers. 

Show that $\sum\limits_{k=0}^\infty z_k$ converges absolutely if and only if  both  $\sum\limits_{k=0}^\infty \mathrm{Re}(z_k)$ and  $\sum\limits_{k=0}^\infty \mathrm{Im}(z_k)$ converge absolutely.

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