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{ \large \textbf{\textsc{Math 5587 -- Homework 12 (Due Thursday Dec 15)}}}
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\begin{enumerate}
\item Find the Green's function for the tilted half-space 
\[D = \{(x,y,z) \, : \, ax + by + cz > 0\}.\]
[Hint: Use the Green's function for the halfspace $\{z>0\}$ and a change of variables.]
\item Find the Green's function for the half ball
\[D = \{(x,y,z) \, : \, x^2 + y^2 + z^2 < a^2 \ \text{ and } \ z>0\}.\]
[Hint: Reflect the solution for the whole ball across the plane $z=0$.]
\item Consider the two dimensional disk
\[D = \{(x,y) \, : \, x^2 + y^2 < a^2\}.\]
Show that the Green's function for the disk is
\[G(\x,\x_0) = \frac{1}{2\pi} \log(\|\x-\x_0\|) - \frac{1}{2\pi} \log\left(\frac{\|\x_0\|}{a}\|\x-\x_0^*\|\right).\]
where $\x_0^* = \frac{a^2\x_0}{\|\x_0\|^2}$. 
\item Use problem 3 to recover the two dimensional version of Poisson's formula for the ball that we derived in class using separation of variables.
\end{enumerate}


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