How to prove that $\int_{-∞}^∞ \int_{-∞}^∞ \operatorname{sinc}(\sqrt{x^2+y^2+z^2})\,dy\,dz=2\pi\cos(x)$? From physical intuition I've found that the following equation should be true
$$\int\limits_{-\infty}^\infty \int\limits_{-\infty}^\infty \operatorname{sinc}\left(\sqrt{x^2+y^2+z^2}\right)\,\mathrm{d}y\,\mathrm{d}z=2\pi\cos(x).$$
Here $$\operatorname{sinc}(x)=\frac{\sin x}x.$$
The intuition is that a plane made of continuously distributed sources of spherical waves should effectively emit plane waves. The proportionality constant $2\pi$ was just guessed based on numerical estimation.
Numerically this identity appears to be true, but how can I prove it?
 A: If we change the $yz$-plane to polar coordinates, the integral is$$2\pi\int_0^\infty\frac{r}{\sqrt{x^2+r^2}}\sin\sqrt{x^2+r^2}dr=2\pi\left[-\cos\sqrt{x^2+r^2}\right]_0^\infty.$$This doesn't converge, because $\lim_{t\to\infty}\cos t$ doesn't exist. Your hypothesis is equivalent to claiming this limit is $0$.
A: From the answer by J.G. we know that, if taken directly, the integral diverges. But there's still a way to recover physical sense from it. Let the medium have an extinction coefficient $\beta>0$. Then the integral will be
$$L=\int\limits_{-\infty}^\infty \int\limits_{-\infty}^\infty \operatorname{sinc}\left(\sqrt{x^2+y^2+z^2}\right)\exp\left(-\beta\sqrt{x^2+y^2+z^2}\right)\,\mathrm{d}y\,\mathrm{d}z.$$
Switching to polar coordinates in the $yz$ plane as done in the above linked answer, and then doing substitution $u=\sqrt{x^2+r^2}$, we'll get the following integral:
$$L=2\pi\int\limits_x^\infty \sin u\exp(-\beta u)\,\mathrm{d}u=
\left.-2\pi\frac{e^{-\beta u}(\cos u+\beta\sin u)}{\beta^2+1}\right|_x^\infty=
2\pi\frac{e^{-\beta x}(\cos x+\beta\sin x)}{\beta^2+1}.$$
Now gradually "idealizing" the system, i.e. removing the extinction, we get
$$\lim_{\beta\to0} L=2\pi\cos x,$$
which is exactly what is claimed in the OP.
