On Integrable Delta Functions on the Levi-Civita Field

Abstract

In this paper, we develop a theory of integrable delta functions on the Levi-Civita field R as well as on R² and R³ with similar properties to the one-dimensional, two-dimensional and three-dimensional Dirac Delta functions and which reduce to them when restricted to points in R, R² and R³, respectively. First we review the recently developed Lebesgue-like measure and integration theory over R, R² and R³. Then we introduce delta functions on R, R² and R³ that are integrable in the context of the aforementioned integration theory; and we study their properties and some applications.