A concise introduction to Colombeau generalized functions and their applications in classical electrodynamics

Abstract

The objective of this introduction to Colombeau algebras of generalized functions (in which distributions can be freely multiplied) is to explain in elementary terms the essential concepts necessary for their application to basic nonlinear problems in classical physics. Examples are given in hydrodynamics and electrodynamics. The problem of the self-energy of a point electric charge is worked out in detail: the Coulomb potential and field are defined as Colombeau generalized functions, and integrals of nonlinear expressions corresponding to products of distributions (such as the square of the Coulomb field and the square of the delta function) are calculated. Finally, the methods introduced in Gsponer (2007 Eur. J. Phys. 28 267, 2007 Eur. J. Phys. 28 1021 and 2007 Eur. J. Phys. 28 1241), to deal with point-like singularities in classical electrodynamics are confirmed.