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

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

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

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

4. M. A. Parseval, Mémoire sur les séries et sur l'intégration complète (etc.) (1805)

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

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

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

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

4. M. A. Parseval, Mémoire sur les séries et sur l'intégration complète (etc.) (1805)

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. 202].

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

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

2. W. F. Donoghue, Distributions and Fourier transforms, Academic Press (1969).

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

4. M. A. Parseval, Mémoire sur les séries et sur l'intégration complète (etc.) (1805)

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

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

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

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

4. M. A. Parseval, Mémoire sur les séries et sur l'intégration complète (etc.) (1805)

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. 202] or [3, p. 428].

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

According to Watson [4, 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 [5], but I am mainly curious about the case n=3, $$ \frac1{4\pi}\int_{S^2}e^{i\langle\mathbf a,\mathbf u\rangle}d\sigma(\mathbf u)=\frac{\sin\|\mathbf a\|}{\|\mathbf a\|}. \tag3 $$

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

2. W. F. Donoghue, Distributions and Fourier transforms, Academic Press (1969).

3. L. Grafakos, Classical Fourier analysis, Springer (2008).

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

5. M. A. Parseval, Mémoire sur les séries et sur l'intégration complète (etc.) (1805)

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. 202].

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

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

2. W. F. Donoghue, Distributions and Fourier transforms, Academic Press (1969).

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

4. M. A. Parseval, Mémoire sur les séries et sur l'intégration complète (etc.) (1805)