Bipolar and Toroidal Harmonics

Laplace’s equation occurs frequently in mathematical physics for problems relating to fluid flow, heat transfer, and so on. For some simple cases, the boundary-value problem can be solved; but more often, the differential equation proves intractable, and numerical analysis or experimental methods are used. The electrooptic analog is an experimental method based upon the fact that an organic dye solution becomes birefringent in an electric field. This effect enables one to determine voltage gradient throughout a two-dimensional field. The boundary conditions most readily applied are prescribed constant values of the electric potential φ on conducting segments of a boundary and ∂φ/∂n = 0 on insulated segments of a boundary. With the known conditions on the boundary and the potential gradient found from experiment, the problem is solved. This analog can be used for all physical problems for which the boundary conditions are applicable, and which satisfy Laplace’s equation.

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V.—On Whittaker's Solution of Laplace's Equation

Proceedings of the Royal Society of Edinburgh Section A Mathematical and Physical Sciences ◽

10.1017/s0080454100006397 ◽

1944 ◽

Vol 62 (1) ◽

pp. 31-36

Author(s):

E. T. Copson

Keyword(s):

Harmonic Function ◽

General Solution ◽

Arbitrary Function ◽

Harmonic Functions ◽

Singular Points ◽

Laplace’S Equation ◽

Laplace's Equation ◽

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In 1902, Professor E. T. Whittaker gave a general solution of Laplace's equation in the formwhere f is an arbitrary function of the two variables. It appears that this is not the most general solution, since there are harmonic functions, such as r−1Q0(cos θ), which cannot be expressed in this form near the origin. The difficulty is naturally connected with the location of the singular points of the harmonic function. It seems therefore to be worth while considering afresh the conditions under which Whittaker's solution is valid.

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Solutions of Laplace's equation in an n-dimensional space of constant curvature

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s001309150000852x ◽

1939 ◽

Vol 6 (1) ◽

pp. 24-45 ◽

Cited By ~ 1

Author(s):

H. S. Ruse

Keyword(s):

Dimensional Space ◽

Flat Space ◽

Christoffel Symbol ◽

Laplace’S Equation ◽

Laplace's Equation ◽

Space Of Constant Curvature ◽

Tensor Formulation ◽

Classical Wave Equation ◽

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Functions of three variables which satisfy both the heat equation and Laplace's equation in two variables

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700039021 ◽

1963 ◽

Vol 3 (4) ◽

pp. 396-407 ◽

Cited By ~ 3

Author(s):

D. V. Widder

Keyword(s):

Differential Equations ◽

Heat Equation ◽

Fluid Mechanics ◽

Laplace’S Equation ◽

Laplace's Equation ◽

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In a recent paper on statistical fluid mechanics Professor J. Kampé de Fériet [1] employed several integrals of which the following is a typical exampleThe functionu(x, y, t), which it defines, formally satisfies the following three classical differential equations

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Note on Whittaker's Solution of Laplace's Equation

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500034313 ◽

1925 ◽

Vol 44 ◽

pp. 22-25

Author(s):

E. T. Copson

Keyword(s):

General Solution ◽

Arbitrary Function ◽

Regular Point ◽

Laplace’S Equation ◽

Laplace's Equation ◽

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§1. Whittaker has shewn that a general solution of Laplace's Equationmay be put in the formwhere f (v, u) denotes an arbitrary function of the two variables u and v; such a representation is valid only in the neighbourhood of a regular point.

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An Integral Formula For Qn (cos θ)

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500024342 ◽

1945 ◽

Vol 7 (2) ◽

pp. 81-82

Author(s):

E. T. Copson

Keyword(s):

General Solution ◽

Present Note ◽

Integral Formula ◽

Simple Expression ◽

Laplace’S Equation ◽

Laplace's Equation ◽

Particular Solution ◽

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The functionis, as is well-known, a general solution of Laplace's equation of degree −1 in (x, y, z). In 1926* I proved that the particular solution r−1Q0 (z/r) cannot be represented in this form whereas the solution r−1Q0 (y/r) can. In the present note I find a very simple expression for the latter solution in the form (1.1), and I deduce from it an apparently new integral formula for Qn (cos θ).

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On Spheroidal Harmonics

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500002364 ◽

1914 ◽

Vol 33 ◽

pp. 65-68

Author(s):

Edward Blades

Keyword(s):

The Other ◽

Laplace’S Equation ◽

Laplace's Equation ◽

Ellipsoidal Harmonics ◽

Spheroidal Harmonics ◽

Principal Axes ◽

The Family ◽

Rectangular Coordinates ◽

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Ellipsoidal harmonics are defined to be those solutions of Laplace's equation(where x, y, z are rectangular coordinates) which are useful in problems relating to ellipsoids. If the equationrepresents a family of confocal quadrics, it is known that the ellipsoidal harmonics belonging to the family are products of the formwhere l1, l2… are constants: one term is to be picked out of the square brackets as a multiplier of the other factors. Now if we consider the case in which two of the principal axes of the ellipsoids are equal, the latter become spheroids. If then we put b = 0 in (1) the family of confocal spheroids has the equationand belonging to this family there will be spheroidal harmonics of the form given by (2) with b zero.

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On Spheroidal Harmonics and Allied Functions

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s001309150000242x ◽

1914 ◽

Vol 33 ◽

pp. 118-121 ◽

Cited By ~ 2

Author(s):

G. B. Jeffery

Keyword(s):

General Solution ◽

General Formula ◽

Laplace’S Equation ◽

Laplace's Equation ◽

Spheroidal Harmonics ◽

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Spheroidal Coordinates

In a paper recently read before this Society, Mr E. Blades obtained a general formula for spheroidal harmonics in the form of the general solution of Laplace's equation given by Professor Whittaker,If spheroidal coordinates r, θ, φ are defined bythe result obtained is

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Transform solutions of Dirichlet problems in quarter-spaces

Glasgow Mathematical Journal ◽

10.1017/s0017089500002330 ◽

1974 ◽

Vol 15 (2) ◽

pp. 156-158

Author(s):

I. N. Sneddon

Keyword(s):

Fourier Transforms ◽

Laplace’S Equation ◽

Dirichlet Problems ◽

Laplace's Equation ◽

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In a recent paper, Boudjelkha and Diaz [1] have considered the solutions of the Dirichlet problems for Laplace's equationand for Helmholtz's equationin¼–ℝn. The purpose of this brief note is to show that their formulae may be derived easily by the use of the theory of multiple Fourier transforms.

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On the “elementary” solution of Laplace's Equation

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500007677 ◽

1931 ◽

Vol 2 (3) ◽

pp. 135-139 ◽

Cited By ~ 4

Author(s):

H. S. Ruse

Keyword(s):

Differential Equation ◽

Partial Differential Equation ◽

Linear Partial Differential Equation ◽

Second Order ◽

Elementary Solution ◽

General Linear ◽

Laplace’S Equation ◽

Partial Differential ◽

Laplace's Equation ◽

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Hadamard defines the “elementary solution” of the general linear partial differential equation of the second order, namely(Aik, BiC being functions of the n variables x1, x2, .., xn, which may be regarded as coordinates in a space of n dimensions), to be one of those solutions which are infinite to as low an order as possible at a given point and on every bicharacteristic through that point.

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