International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 100   | 1 | 2 |

Section 1.3.4.5.1.1. Circular harmonic expansions in polar coordinates

G. Bricognea

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1.3.4.5.1.1. Circular harmonic expansions in polar coordinates

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Let [f = f(x, y)] be a reasonably regular function in two-dimensional real space. Going over to polar coordinates[x = r \cos \varphi\quad y = r \sin \varphi]and writing, by slight misuse of notation, [f(r, \varphi)] for [f(r \cos \varphi, r \sin \varphi)] we may use the periodicity of f with respect to ϕ to expand it as a Fourier series (Byerly, 1893[link]):[f(r, \varphi) = {\textstyle\sum\limits_{n \in {\bb Z}}} \,f_{n} (r) \exp (in \varphi)]with[f_{n} (r) = {1 \over 2\pi} {\textstyle\int\limits_{0}^{2\pi}} f(r, \varphi) \exp (-in \varphi) \hbox{ d}\varphi.]

Similarly, in reciprocal space, if [F = F(\xi, \eta)] and if[\xi = R \cos \psi\quad \eta = R \sin \psi]then[F(R, \psi) = {\textstyle\sum\limits_{n \in {\bb Z}}} \,i^{n} F_{n} (R) \exp (in\psi)]with[F_{n} (R) = {1 \over 2\pi i^{n}} {\textstyle\int\limits_{0}^{2\pi}} F(R, \psi) \exp (-in \psi) \hbox{ d}\psi,]where the phase factor [i^{n}] 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.]








































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