Fourier transform of characteristic function in a sphere

Question

A similar question was asked before for an interval in $\mathbb{R}$. I wonder how to do it for a characteristic function of $\{x\in\mathbb{R}^3:|x|<r\}$ i.e. I want to calculate $$ \frac{1}{(2\pi)^{\frac{3}{2}}}\int_{\mathbb{R^3}} \chi_{|x|<r}(x)\exp(-ikx) dx.$$ Could you please help me?

Answer 1

Choosing the $z$ axis along $k$ and denoting $|k|$ by $q$, we have

$$ \begin{align} \int_{\mathbb R^3}\chi_{|x|\lt r}(x)\exp(-\mathrm ikx)\,\mathrm dx &=\int_0^r R^2\,\mathrm dR\int_0^\pi\sin\theta\,\mathrm d\theta\int_0^{2\pi}\mathrm d\phi\exp(-\mathrm iqR\cos\theta) \\ &=2\pi\int_0^r R^2\,\mathrm dR\int_0^\pi\sin\theta\,\mathrm d\theta\exp(-\mathrm iqR\cos\theta) \\ &=2\pi\int_0^r R^2\,\mathrm dR\left[\frac1{\mathrm iqR}\exp(-\mathrm iqR\cos\theta)\right]_{\theta=0}^{\theta=\pi} \\ &=\frac{2\pi}{\mathrm iq}\int_0^r \mathrm dRR\left(\exp(\mathrm iqR)-\exp(-\mathrm iqR)\right) \\ &=\frac{4\pi}{q^3}\left(r\cos qr-\sin qr\right)\;. \end{align} $$