GENERALIZATION OF THE THOMSON FORMULA FOR HARMONIC FUNCTIONS OF A GENERAL TYPE

It is shown that the Thomson formula for three-dimensional harmonic functions is unique. Namely, there are no other formulas of this type, with the exception of the trivial change of variables in the form of shifts, reflections, rotations and stretching of coordinates. However, the Thomson formula c...

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Published inSt. Petersburg Polytechnical University Journal. Physics and Mathematics Vol. 12; no. 2
Main Authors Berdnikov Alexander, Gall Lidia, Gall Nikolaj, Solovyev Konstantin
Format Journal Article
LanguageEnglish
Published Peter the Great St.Petersburg Polytechnic University 01.06.2019
Subjects
electrostatic field
laplace’s equation
magnetostatic field
scalar potential
thomson formula
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Summary:It is shown that the Thomson formula for three-dimensional harmonic functions is unique. Namely, there are no other formulas of this type, with the exception of the trivial change of variables in the form of shifts, reflections, rotations and stretching of coordinates. However, the Thomson formula can be generalized if, instead of purely algebraic linear expressions, one uses a linear algebraic form with the participation of the first order partial derivatives of the source function. The paper provides an exhaustive list of first order differentiating expressions that convert arbitrary three-dimensional harmonic functions into new three-dimensional harmonic functions. All these formulas obtained for the space of three dimensions can be transferred to the multidimensional case as well.
ISSN:2405-7223
DOI:10.18721/JPM.12203