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 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.

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.

The electrostatics of conductors

2018 ◽  
Author(s):  
J. Pierrus
Keyword(s):  
Harmonic Functions ◽  
Conformal Transformation ◽  
Early Stage ◽  
Fourier Method ◽  
Relaxation Method ◽  
Numerical Calculations ◽  
Laplace’S Equation ◽  
Laplace's Equation ◽  
Electrostatic Equilibrium

This chapter begins by proving some important properties of (i) conductors in electrostatic equilibrium, and (ii) harmonic functions. These results underpin most of the remaining questions of Chapter 3. The coefficients of capacitance for an arbitrary arrangement of conductors are introduced at an early stage, and numerical calculations then follow in a number of subsequent questions. Some important techniques (both analytical and numerical) for finding solutions to Laplace’s equation are considered. These include: the Fourier method, the relaxation method, themethod of images and the method of conformal transformation. All of these are discussed in some detail, and with appropriate examples.

A note on positive harmonic functions

1948 ◽  
Vol 44 (2) ◽  
pp. 289-291 ◽  
Author(s):  
S. Verblunsky
Keyword(s):  
Harmonic Function ◽  
Harmonic Functions ◽  
Duke Math ◽  
Negative Number ◽  
Image Position ◽  
American Math ◽  
Positive Harmonic Functions

If H(ξ, η) is a harmonic function which is defined and positive in η > 0, then there is a non-negative number D and a bounded non-decreasing function G(x) such that(For a proof, see Loomis and Widder, Duke Math. J. 9 (1942), 643–5.) If we writewhere λ > 1, then the equationdefines a harmonic function h which is positive in υ > 0. Hence there is a non-negative number d and a bounded non-decreasing function g(x) such thatThe problem of finding the connexion between the functions G(x) and g(x) has been mentioned by Loomis (Trans. American Math. Soc. 53 (1943), 239–50, 244).

A theorem on positive harmonic functions

1949 ◽  
Vol 45 (2) ◽  
pp. 207-212 ◽  
Author(s):  
S. Verblunsky
Keyword(s):  
Harmonic Function ◽  
Unit Circle ◽  
Harmonic Functions ◽  
Closed Interval ◽  
Positive Harmonic Function ◽  
Image Position ◽  
Function Conjugate ◽  
Positive Harmonic Functions

1. Let z = reiθ, and let h(z) denote a (regular) positive harmonic function in the unit circle r < 1. Then h(r) (1−r) and h(r)/(1 − r) tend to limits as r → 1. The first limit is finite; the second may be infinite. Such properties of h can be obtained in a straightforward way by using the fact that we can writewhere α(phgr) is non-decreasing in the closed interval (− π, π). Another method is to writewhere h* is a harmonic function conjugate to h. Then the functionhas the property | f | < 1 in the unit circle. Such functions have been studied by Julia, Wolff, Carathéodory and others.

Minimum growth of harmonic functions and thinness of a set

1984 ◽  
Vol 95 (1) ◽  
pp. 123-133 ◽  
Author(s):  
Jang-Mei G. Wu
Keyword(s):  
Harmonic Function ◽  
Harmonic Functions ◽  
Integral Condition ◽  
Finite Point ◽  
Image Position ◽  
Increasing Function

In [3], Barth, Brannan and Hayman proved that if u(z) is any non-constant harmonic function in ℝ2, ø(r) is a positive increasing function of r for r ≥ 1 andthen there exists a path going from a finite point to ∞, such that u(z) > ø(|z|) on Γ. Moreover, they showed by example that the integral condition above cannot be relaxed.

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.

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