I am reading the paper Asymptotic conditions and infrared divergences in quantum electrodynamics by P. P. Kulish & L. D. Faddeev (the paper is not important for the question I think, but I will include the link).
I encounter an integral at some point, which I am having a lot of difficulty to prove: the integral is this $$\int\frac{d^3\vec{k}}{2k^0}\sin\bigg[k\cdot\bigg(\frac{q}{q^0}\tau-\frac{p}{p^0}s\bigg)\bigg]=2\pi^2\delta\bigg[\bigg(\frac{q}{q^0}\tau-\frac{p}{p^0}s\bigg)^2\bigg]\varepsilon(\tau-s)$$ where $\varepsilon(x)\equiv{\rm sgn}(x)$. So, how do I prove that this integral holds? If so, how do I perform an integral over $s$ at some later stage since the delta function has a power of two in its argument i.e. what is the integral $\int dx\ \delta[(x-a)^2]f(x)$?
Any help will be appreciated