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)

thatelementary...) Then OK, I replaced "first derived" by "first published". Note that Watson or Encykl. Math. Wiss.attributescores of formulas -- but not this one, as far as I could find. $\endgroup$moduloArchimedes, I'd just be surprised to see Poisson state it (below, item 2.) without at least as much of an argument as you have given --ifthis was the first time it appeared. Instead, he justifies it by calling it a "known formula". To me this suggests there may be an earlier occurrence, and this "first" is what I'm looking for. But maybe it doesn't exist. $\endgroup$6more comments