scholarly journals Note on Whittaker's Solution of Laplace's Equation

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 ◽  
Image Position

§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.

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

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 ◽  
Image Position

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.

scholarly journals An Integral Formula For Qn (cos θ)

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 ◽  
Image Position

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).

scholarly journals On Spheroidal Harmonics and Allied Functions

1914 ◽  
Vol 33 ◽  
pp. 118-121 ◽  
Author(s):  
G. B. Jeffery
Keyword(s):  
General Solution ◽  
General Formula ◽  
Laplace’S Equation ◽  
Laplace's Equation ◽  
Spheroidal Harmonics ◽  
Image Position ◽  
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

scholarly journals Transformations of Axes for Whittaker's Solution of Laplace's Equation

1916 ◽  
Vol 35 ◽  
pp. 32-37 ◽  
Author(s):  
G. B. Jeffery
Keyword(s):  
General Solution ◽  
Laplace’S Equation ◽  
Laplace's Equation ◽  
Rectangular Coordinates ◽  
Image Position

Whittaker has shown that a general solution of Laplace's equation, ∇2V = 0, may be expressed in the formSince the harmonic property of a function is in no way dependent upon any particular set of axes it follows that the same solution must be capable of being expressed in the formwhere X, Y, Z are any second set of rectangular coordinates.

scholarly journals Note on the electric capacity coefficients of spheres

1912 ◽  
Vol 87 (598) ◽  
pp. 485-487
Keyword(s):  
General Solution ◽  
Laplace’S Equation ◽  
Laplace's Equation ◽  
Electric Capacity ◽  
Physical Problems

In the 'Proceedings' of the Society, vol. 87, p. 109, Mr. Jeffery obtains a general solution of Laplace’s equation in a form suitable for physical problems in connection with two spheres. As an illustration he applies his solution to the problem of finding the capacity coefficients of two equal spheres, obtaining a result which he shows to be equivalent to one of Maxwell’s series formulæ. He then computes a table of the numerical values of these coefficients.

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

1939 ◽  
Vol 6 (1) ◽  
pp. 24-45 ◽  
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 ◽  
Image Position ◽  
Similar Solutions

This paper is a sequel to an earlier one containing a tensor formulation and generalisation of well-known solutions of Laplace's equation and of the classical wave-equation. The partial differential equation considered waswhere is the Christoffel symbol of the second kind, and the work was restricted to the case in which the associated line-elementwas that of an n-dimensional flat space. It is shown below that similar solutions exist for any n-dimensional space of constant positive or negative curvature K.

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

1963 ◽  
Vol 3 (4) ◽  
pp. 396-407 ◽  
Author(s):  
D. V. Widder
Keyword(s):  
Differential Equations ◽  
Heat Equation ◽  
Fluid Mechanics ◽  
Laplace’S Equation ◽  
Laplace's Equation ◽  
Image Position

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

scholarly journals On Spheroidal Harmonics

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 ◽  
Image Position

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.

scholarly journals Bipolar and Toroidal Harmonics

1915 ◽  
Vol 34 ◽  
pp. 102-108 ◽  
Author(s):  
G. B. Jeffery
Keyword(s):  
Integral Form ◽  
Mathematical Physics ◽  
Laplace’S Equation ◽  
Laplace's Equation ◽  
Image Position

Most of the solutions of Laplace's equation in common use in mathematical physics have been expressed in the integral form given by Whittaker, viz

scholarly journals VI. Two-dimensional potential problems concerning a single closed boundary

1929 ◽  
Vol 228 (659-669) ◽  
pp. 235-274 ◽  
Keyword(s):  
General Solution ◽  
Boundary Conditions ◽  
Steady Flow ◽  
Applied Mathematics ◽  
Physical Significance ◽  
Two Dimensional ◽  
Laplace’S Equation ◽  
Boundary Values ◽  
Laplace's Equation ◽  
Potential Problems

Solutions of Laplace’s equation, ∂ 2 V/∂ x 2 + ∂ 2 V/∂ y 2 + ∂ 2 V/∂ z 2 = 0 . . . . . (1. 11) are required in many branches of Applied Mathematics, such as hydrodynamics, electro-and magneto-statics, steady flow of heat or electricity, etc. The two-dimensional form of the equation, ∂ 2 V/∂ x 2 + ∂ 2 V/∂ y 2 = 0, . . . . (1. 12) has a general solution V = f ( x + ɩy ) + F ( x – ɩy ), . . . (1. 21) f and F being arbitrary functions of their complex arguments. In the applications, one function alone is usually sufficient, and it is customary to write w = ϕ + ɩψ = f ( z ). . . . . (1. 22) with z = x + ɩ y , when ϕ and ψ usually have each some physical significance. Moreover, in most cases, the boundary conditions which have to be satisfied either are, or can be reduced to, the prescription of the boundary values of ϕ or ψ, of their derivatives.

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