# The integral of $\exp(-|x-a|)$ over an even dimensional sphere

I'm after a reference for an integral. For $m$ a positive integer and $R>0$ let $S^{2m}_R\subset \mathbb{R}^{2m+1}$ denote the radius $R$ sphere of dimension $2m$. Suppose that $a$ lies inside the sphere, i.e. $|a|<R$. I wish to know the following integral as a function of $R$ and $|a|$. $$\int_{S^{2m}_R} e^{-|x-a|} \ \ \mathrm{d}x$$ I have a formula essentially in terms of $e^R$, $e^{|a|}$ and reverse Bessel polynomials in $R$ and in $|a|$. Sage can verify my formula up to $m=20$ (and beyond).

I haven't yet figured out how to prove the formula, but think that this must be a classical computation. Can anyone provide me with a reference to this in the literature?

[Edit: I have now asked a new question Proof of an Identity Involving an Integral over an Even Sphere giving the formula explicitly and asking for a proof. ]

• isn't the calculation just a matter of writing $dx$ in hyperspherical coordinates and carrying out one radial integration and one angular integration? – Carlo Beenakker May 23 '16 at 6:38
• @CarloBeenakker Yes. That's what how you can calculate the integral and that's how I get Sage to calculate the integral. However, what I want is a closed form expression in terms of the reverse Bessel polynomials. As I say what I'm after here is whether or not such a formula has appeared in the literature before. – Simon Willerton May 23 '16 at 7:48