Solving a 2 variable integral with a delta function How can this integrals that include the dirac delta function be solved?
$$\int_{-\infty}^{+\infty}dq\int_{-\infty}^{+\infty}dp\cdot p^n\cdot \delta(p^2+q^2-E)$$

Note that the the integral is zero by symmetry if $n$ is odd. Assume from now on that $n\geq 0$ is even.
One idea is to use polar coordinates $$(p,q)~=~(r\cos\theta,r\sin\theta).\tag{1}$$
Then
$$I~:=~ \iint_{\mathbb{R}^2}\! \mathrm{d}p~\mathrm{d}q~p^n~\delta(p^2+q^2-E)
~=~ \int_{\mathbb{R}_+}\! \mathrm{d}r~r^{n+1}\delta(r^2-E)\int_{[0,2\pi]}\! \mathrm{d}\theta~\cos^n\theta $$
$$~=~\ldots~=~\frac{1}{2}H(E) |E|^{n/2} 2\pi \frac{(n-1)!!}{n!!}.\tag{2}$$
