Suppose a function in spherical coordinates $f(r,\phi,\theta)$. Ask for the Fourier transformation of such function.
Suppose that the function $f$ was symmetric with respect $\phi,\theta$, i.e. $f(r,\phi,\theta)=f(\theta)$. Then what's the spherical coordinates Fourier transformation for such function?
Because, the usual definition $$\int_{0}^\infty f(r,\phi,\theta) r^2dr \int_0^{2\pi}d\phi \int_0^\pi d\theta e^{-i\vec{p}\cdot \vec{x}}$$ didn't seem to quite apply. Since $\vec{x}=(r,\phi,\theta)$ and $\vec{p}=(p_r,p_\phi,p_\theta)$, the latter doesn't even seemed to be properly defined. Even if one treat it as a one dimensional cases, will the coefficient be $1/\sqrt{2\pi}^3$ or just $1/\sqrt{2\pi}$?
How to do the fourier transformation in spherical coordinates?