International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. B. ch. 1.3, p. 93
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Let be a reasonably regular function in two-dimensional real space. Going over to polar coordinates and writing, by slight misuse of notation, for we may use the periodicity of f with respect to φ to expand it as a Fourier series (Byerly, 1893): with
Similarly, in reciprocal space, if and if then with where the phase factor has been introduced for convenience in the forthcoming step.
References
Byerly, W. E. (1893). An elementary treatise on Fourier's series and spherical, cylindrical and ellipsoidal harmonics. Boston: Ginn & Co. [Reprinted by Dover Publications, New York, 1959.]Google Scholar