XII.—Paraboloidal Co-ordinates and Laplace's Equation
1963 ◽
Vol 66
(3)
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pp. 129-139
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SynopsisIn this paper we examine the general paraboloidal co-ordinate system, in which the normal surfaces are elliptic or hyperbolic paraboloids, including as special cases the “parabolic plate” and the “plate with a parabolic hole”. We then show that normal solutions of Laplace's equation in these co-ordinates are given as products of three Mathieu functions, and apply this to the solution of boundary-value problems for Laplace's equation in these co-ordinates. In a subsequent paper the corresponding treatment of the wave equation will be given.
2013 ◽
Vol 2013.26
(0)
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pp. _2008-1_-_2008-3_
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2004 ◽
pp. 121-134
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