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\begin{document}

\section*{Problem Set 5}

%Hand in all.
\begin{enumerate}
\item Prove that there does not exist a Green's function for the region $\CC$. (You may take the point $z_0$ to be $0$.)
\item (a) Find the Green's function $g(z,z_0)$ for the upper half-plane $\mathfrak{H}$ (with singularity at $z_o\in \mathfrak{H}$), and show it extends to a neighborhood of $\mathfrak{H}\cup \RR$.  (b) Given a continuous, real-valued function $f$ on $\RR$ with $\lim_{x\to +\infty}f(x)=\lim_{x\to -\infty}f(x)<\infty$, show that\footnote{The meaning of $\tfrac{\partial g}{\partial y}(x,z_0)$ is that we are taking the vertical partial derivative in the first variable $z$ and evaluating at $z=x+i0$. We are now thinking of $z_0$ as a \emph{variable} point in $\mathfrak{H}$.} $u(z_0):=\int_{\R}\tfrac{\partial g}{\partial y}(x,z_0)f(x)dx$ solves the Dirichlet problem on $\mathfrak{H}$ with $f$ as boundary data.
\item Show that the period matrix for multiply connected regions (defined in my notes, or in Ahlfors) is symmetric.
\item Let $\Omega$ be the doubly connected region obtained by removing the vertical strips $[ia,ib]$ and $[\mu+ic,\mu+id]$ from $\CC$. This is conformally equivalent to an annulus $A(1,\lambda)$ for a unique $\lambda>1$. (a) Determine $\lambda$. (b) Can you relate this to a complex torus $\CC/\Lambda$? (Actually there is more than one way to do this.)
\end{enumerate}

\end{document}
