Hello!


It may be a stupid question, i'm trying to find a closed form for an integral similar to a Fourier transform on $S^{n}$ but i'm stuck...
Let $\alpha>0$, the integral i can't solve is


`$$I(p,\alpha)=\int_{S^{n}}e^{i\alpha\left<p,q\right>}d\mu_{S^{n}}(q)$$`


where $p,q\in S^{n}$, $\left< \cdot,\cdot \right>$ is the euclidean scalar product on $\mathbb{R}^{n+1}$ and $d\mu\_{S^{n}}$ is the measure induced by euclidean measure on $\mathbb{R}^{n+1}$. so the question is: is there a closed form for $I(p,\alpha)$ only in terms of $p$ and $\alpha$? 


Thank you in advance!



Closed form for Fourier transform-like Integral on $S^{n}$
The integral obviously doesn't depend on $p$, since it must be rotationally invariant, and, using polar coordinates centered on $p$, one obtains
$$
I(p,\alpha) = \mathrm{vol}(S^{n-1})\int\_{-1}^1 e^{i\alpha t}(1{-}t^2)^{(n-2)/2}dt
=\frac{2\ \pi^{n/2}}{\Gamma(n/2)}\int\_{-1}^1 \cos(\alpha t)(1{-}t^2)^{(n-2)/2}dt.
$$
This latter integral can be evaluated by standard techniques. For odd values of $n$, it is expressed in Bessel functions, while, for even values of $n$ it is expressed in terms of elementary functions. Thus, for $n=2$, one has
$$
\int\_{-1}^1 \cos(\alpha t)(1{-}t^2)^{(2-2)/2}dt = \frac{2\sin\alpha}{\alpha},
$$
while, for $n=3$, one has
$$
\int\_{-1}^1 \cos(\alpha t)(1{-}t^2)^{(3-2)/2}dt = \frac{\pi\ J\_1(\alpha)}{\alpha},
$$
where $J\_1$ denotes the Bessel function of the first kind, and so on.


