Distributional jump formula for a Heaviside product ($D^k(Hu)=H{u}^{(k)}+{\sum }_{s=0}^{k-1}{u}^{(k-1-s)}(0){\delta }_0^{(s)}$)

For a smooth function $u$, the distributional derivative of the product of $u$ with the Heaviside function is $D(Hu)=H{u}^{\prime }+u(0){\delta }_0$. Induction gives the displayed formula. It records all boundary sources, including derivatives of the Dirac delta distribution, and determines which initial derivatives must vanish in a retarded fundamental solution.