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Potential functions with periodicity in one coordinate

Published online by Cambridge University Press:  24 October 2008

R. C. J. Howland
R. C. J. Howland
Affiliation:
Emmanuel College
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Extract

The general conception underlying the following analysis is that of a field of potential which is invariant under a group of geometrical transformations. The boundary is to consist of a number of parts which transform into each other and the boundary values also transform into each other. To satisfy the conditions we build up functions which are invariant under the same group of transformations and combine them to give the prescribed boundary values over one section of the boundary. The boundary conditions on the other sections are then automatically satisfied. When the groups are simply translations or rotations we get functions periodic in either a Cartesian or an angular coordinate.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 30 , Issue 3 , July 1934 , pp. 315 - 326
Copyright
Copyright © Cambridge Philosophical Society 1934

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References

REFERENCES

(1)Knight, R. C., “The potential of a circular cylinder between infinite planes”, Proc. Lond. Math. Soc. (to be published shortly).Google Scholar
(2)Hobson, E. W., Spherical and Ellipsoidal Harmonics (Cambridge, 1931), pp. 135 and 139.Google Scholar
(3)Rigby, C. M., “The electric field of a sphere lying between two infinite conducting planes”, Proc. Lond. Math. Soc. (2), 33 (1932), 525–530.CrossRefGoogle Scholar