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I'd like to know how the following integral is derived (or, at least, a reference detailing how it is done, or a table of integrals which contains this). For any $\rho > 0$ and non-zero vector $b \in \mathbb{R}^n$, we have the following multidimensional integral over $k \in \mathbb{R}^n$:

$$\displaystyle \int_{|k| < \rho}e^{ibk} \ \mathrm{d}k = \left( \frac{2\pi \rho}{|b|}\right)^{n/2}J_{n/2}(\rho |b|),$$

where the term $bk$ denotes the scalar product of the vectors $b, k$.

Numerical evidence shows that the formula holds true for $n = 1,2$. But how is this identity actually proved? The only thing I can think of is that you could obtain an expression like this by integrating the Taylor series of the exponential function and then linking that to the known series expansions of $J_{\alpha}$. There is also a similar integral here, but it isn't quite the same as mine.

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    You just need to integrate along "shells" to transform the LHS into the definition of a Bessel function. – 2017-01-18
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    choose $bk=|k||b|\cos(\phi)$. Now use spherical coordinates to get a trivial $\|k|$ integral. the angular integrals are all trivial expect of one which gives the desired Bessel function – 2017-01-18
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    Or: multiply both sides by $\rho^{-n/2}$ then prove the LHS and the RHS fulfill the same differential equation (Bessel's differential equation) with respect to the $\rho$ variable. – 2017-01-18

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