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, 
$$
\frac1{4\pi}\int\_{S^2}e^{i\langle\mathbf a,\mathbf u\rangle}d\sigma(\mathbf u)=\frac{\sin\|\mathbf a\|}{\|\mathbf a\|}.
\tag3
$$




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1. I. M. Gel'fand & G. E. Shilov, [Generalized functions, vol. 1](http://www.ams.org/mathscinet-getitem?mr=166596), Academic Press (1964).
2. E. M. Stein & G. Weiss, [Introduction to Fourier analysis on Euclidean spaces](http://books.google.com/books?id=YUCV678MNAIC&pg=PA154), Princeton UP (1971).
3. G. N. Watson, [A treatise on the theory of Bessel functions](https://archive.org/stream/treatiseontheory00watsuoft#page/9), Cambridge UP (1922)
4. M. A. Parseval, [Mémoire sur les séries et sur l'intégration complète (etc.)](http://books.google.com/books?id=D1s3AQAAMAAJ&pg=PA638) (1805)



Fourier transform of the unit sphere
At the risk of answering my own question, here is what I have since found:


1. For general $n$, formula (1) seems to occur first on p. 177 of S. Bochner, [*Summation of multiple Fourier series by spherical means*](http://www.ams.org/mathscinet-getitem?mr=1501870), Trans. AMS **40** (1936) 175-207. Bochner exposes it again on pp. 73-74 of [*Fourier Transforms*](http://www.ams.org/mathscinet-getitem?mr=31582) (Princeton UP 1949).
2. For $n=3$, Burkhardt (*Trigonometrische Reihen und Integrale bis etwa 1850*, Encykl. Math. Wiss. **II A 12** (1916) 819-1354, [page 1258](https://archive.org/stream/p1n2encyklopdied02akaduoft#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](http://books.google.com/books?id=TZ8AAAAAYAAJ&pg=PA134), 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](http://books.google.com/books?id=l0dUAAAAcAAJ&pg=PA558)). Perhaps someone will have better luck locating the original (3) -- in Poisson or elsewhere?




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**Edit:** Aha, the problem was simply a typo in Burkhardt. Formula (3) indeed appears in Poisson's above-cited *Mémoire*, but on [page 174](http://books.google.com/books?id=TZ8AAAAAYAAJ&pg=PA174) instead of 134, in the form
$$
\int\int e^{at(g\cos u+h\sin u\sin v+k\sin u\cos v)}\sin u\,du\,dv =
2\pi\frac{e^{atp}-e^{-atp}}{atp}.
$$


