Math 6420 • PDE

Let \(x_+=xH(x)\), where \(x\in\mathbb{R}\) and \(H(x)\) is the Heaviside function so that
\(x_+=x\) for \(x>0\) and \(x_+=0\) for \(x\le0\).
\begin{enumerate}
\item[(a)] Show that differentiating in the sense of distributions we have for any integer
\(k\ge0\) that
\[
  \partial(x_+^k)=
  \begin{cases}
    kx_+^{k-1}, & \text{if } k\ge1,\\
    \delta, & \text{if } k=0.
  \end{cases}
\]

\item[(b)] Let \(k\ge1\) be an integer. Use (a) to find a distribution \(E\) such that
\(\partial^kE=\delta\).
\end{enumerate}