On the multiplicative product of the Dirac-delta distribution on the hyper-surface (Q2714012)

Let \(x=(x_1,\dots ,x_n)\) be a point in the Euclidean space \(\mathbb{R}^n\). Put \(r=\sqrt {x_1^2+\dots +x_p^2}\), \(s=\sqrt {x_{p+1}^2+\dots +x_{p+q}^2}\) (\(p+q=n\)). The author defines the generalized functions \(\frac{\delta (cr-s)}{r^{\frac{p-1}{2}}s^{\frac{q-1}{2}}}\) and \(\frac{\delta (cr+s)}{r^{\frac{p-1}{2}}s^{\frac{q-1}{2}}}\) for some \(c\in \mathbb{R}\). He also defines the multiplicative product of such functions and proves the formula \(\frac{\delta (cr-s)}{r^{\frac{p-1}{2}}s^{\frac{q-1}{2}}} \frac{\delta (cr+s)}{r^{\frac{p-1}{2}}s^{\frac{q-1}{2}}}=\frac{c^{p-1} \pi ^{\frac{n}{2}}\delta (x)}{\Gamma(\frac{p}{2})\Gamma(\frac{q}{2})}\). In the case \(n=4\) and \(p=1\) this formula is used for perturbative calculation of the Green function in quantum field theories.