International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2010 
International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 93

Green's theorem stated in terms of distributions (Section 1.3.2.3.9.1) is particularly well suited to the calculation of the Fourier transforms of indicator functions. Let f be the indicator function and let S be the boundary of U (assumed to be a smooth surface). The jump in the value of f across S along the outer normal vector is , the jump in the normal derivative of f across S is , and the Laplacian of f as a function is (almost everywhere) 0 so that . Green's theorem then reads:
The function satisfies the identity . Therefore, in Cartesian coordinates:i.e.where n is the outer normal to S. This formula was used by von Laue (1936) for a different purpose, namely to calculate the transforms of crystal shapes (see also Ewald, 1940). If the surface S is given by a triangulation, the surface integral becomes a sum over all faces, since n is constant on each face. If U is a solid sphere with radius R, an integration by parts gives immediately:
References
Ewald, P. P. (1940). Xray diffraction by finite and imperfect crystal lattices. Proc. Phys. Soc. London, 52, 167–174.Laue, M. von (1936). Die aüssere Form der Kristalle in ihrem Einfluss auf die Interferenzerscheinungen an Raumgittern. Ann. Phys. (Leipzig), 26, 55–68.