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

Abstract

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.

Keywords

• electrostatic field
• magnetostatic field
• scalar potential
• laplace’s equation
• thomson formula