An introduction to generalized functions and their application to static electromagnetic point dipoles, including hyperfine interactions

The Dirac δ function, the Heaviside step function, and all their derivatives are examples of generalized functions, also called distributions. The mathematical theory of generalized functions provides a useful framework for the description of the potentials and fields occurring in electromagnetic theory. It allows for greater clarity and applicability than that given by ordinary functions, especially where point charges and point dipoles are concerned. The present article provides a logical tutorial presentation of the relevant material. Maxwell’s equations for static point sources are reinterpreted in terms of potentials and fields that are generalized functions. Dipolar interactions are discussed in terms of these. Specifically, as an example, the hyperfine spin Hamiltonian for a 1‐electron system is considered in some detail.