TITLE: Fourier transform of the unit sphere
QUESTION [22 upvotes]: 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)

REPLY [2 votes]: 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.