Math 6410 • ODE

\textbf{Distributions.}
Let \(k(x,y)=\frac{1}{2\pi}\log\lvert x-y\rvert\), where \(x,y\in\R^2\).
\begin{enumerate}
\item[(a.)] Show that \(k(\cdot,y)\) is a distribution on \(\R^2\) for each \(y\in\R^2\).
\item[(b.)] Show that as distributions, \(\Delta_x k(x,y)=\delta_y(x)\) for all
\(y\in\R^2\). [\(\delta_y(x)=\delta(x-y)\) and \(\delta\) is the Dirac delta function.]
\end{enumerate}