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 be a reasonably regular function in two-dimensional real space. Going over to polar coordinatesand 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 ifthenwithwhere 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.]