Fourier transform of the unit sphere The Fourier transform of the volume form of the (n-1)-sphere in $\mathbf R^n$ is given by the well-known formula
$$
\int_{S^{n-1}}e^{i\langle\mathbf a,\mathbf u\rangle}d\sigma(\mathbf u) = (2\pi)^{\nu + 1}\|\mathbf a\|^{-\nu}J_\nu(\|\mathbf a\|),
\qquad\nu=\frac n2 -1,
\tag1
$$
found e.g. in [1, p. 198] or [2, p. 154].

Does anyone here know earlier references, and perhaps who first published this formula?

According to Watson [3, p. 9] the case n=2,
$$
\frac1{2\pi}\int_{S^1}e^{ia\cos\theta}d\theta=J_0(a)
\tag2
$$
goes back to Parseval [4], but I am mainly curious about the case n=3, 
$$
\frac1{4\pi}\int_{S^2}e^{i\langle\mathbf a,\mathbf u\rangle}d\sigma(\mathbf u)=\frac{\sin\|\mathbf a\|}{\|\mathbf a\|}.
\tag3
$$



*

*I. M. Gel'fand & G. E. Shilov, Generalized functions, vol. 1, Academic Press (1964). 

*E. M. Stein & G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton UP (1971).

*G. N. Watson, A treatise on the theory of Bessel functions, Cambridge UP (1922)

*M. A. Parseval, Mémoire sur les séries et sur l'intégration complète (etc.) (1805) 
 A: Sonin computed (according to Fichtenholz, but no reference given) $$\int\limits_{\sum_{k=1}^n x_k^2\leq 1}\exp(\langle a,x\rangle) dx_1...dx_n.$$ Fichtenholz did research in multivariate integration, so he knew all these things from folklore, I guess.

A: At the risk of answering my own question, here is what I have since found:


*

*For general $n$, formula (1) seems to occur first on p. 177 of S. Bochner, Summation of multiple Fourier series by spherical means, Trans. AMS 40 (1936) 175-207. Bochner exposes it again on pp. 73-74 of Fourier Transforms (Princeton UP 1949).

*For $n=3$, Burkhardt (Trigonometrische Reihen und Integrale bis etwa 1850, Encykl. Math. Wiss. II A 12 (1916) 819-1354, page 1258) claims to find formula (3) in Poisson's Mémoire sur l'intégration de quelques équations linéaires aux différences partielles, et particulièrement de l'équation générale du mouvement des fluides élastiques, Mém. Acad. Roy. Sci. Inst. France 3 (1820) 121-176, page 134, in the form
$$
\mathfrak{Sin}\,pt=
\frac{pt}{2\pi}\int_0^{2\pi}\int_0^\pi\exp\{t(g\cos u+h\sin u\sin v+k\sin u\cos v)\}\sin u\,du\,dv
$$
where $p=\sqrt{\smash[b]{g^2+h^2+k^2}}$, $\mathfrak{Sin}$ is a hyperbolic function, and Burkhardt is missing a factor of 2. However... I'm not able to find it on that page of Poisson. On the other hand Poisson states it as "known" in a later memoir (1831, page 558). Perhaps someone will have better luck locating the original (3) -- in Poisson or elsewhere?

Edit: Aha, the problem was simply a typo in Burkhardt. Formula (3) indeed appears in Poisson's above-cited Mémoire, but on page 174 instead of 134, in the form
$$
\int\int e^{at(g\cos u+h\sin u\sin v+k\sin u\cos v)}\sin u\,du\,dv =
2\pi\frac{e^{atp}-e^{-atp}}{atp}.
$$
A: The answer is:
Weyl, H. 1919, Annalen der Physik, 365, 481
doi: 10.1002/andp.19193652104
Though it's difficult to find in there if you don't understand German
