How to calculate this surface integral

Question

In a calculation about a Fourier transform, I stumbled in the following surface integral. Let $\mathcal{S}_r$ be the sphere of radius $r>0$ centered in $0$ in $\mathbb{R}^n$ and let $\mathcal{H}^{n-1}$ be the $n-1$ dimensional Hausdorff measure in $\mathbb{R}^n$. If $x\in\mathbb{R}^n$, I've no idea how to explicitly calculate: $$\int_{S_r}e^{-ix\cdot \xi}d\mathcal{H}^{n-1}(\xi).$$ It is even possible to give an explicit formula? Thanks in advance.

Answer 1

The Fourier transform of the unit sphere is expressed in terms of a Bessel function: see Fourier transform of the unit sphere on MathOverflow. It is also briefly discussed on this site, in the question About the Fourier transform of the surface measure of the unit sphere.