Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 327 / 2 / c / Solution

Factor the operator asSet $f(x)=H(x)\sin x$ and $g(x)=H(x)\sinh x$. Since both vanish at zero and have right derivative one, their distributional second derivatives areBecause $D^2=-{\partial }^2$,Consequently $E(x_1,x_2)=f(x_1)g(x_2)$ obeysIt equals $\sin x_1\sinh x_2$ in the positive quadrant and zero otherwise.