On the radial derivative of the delta distribution

TL;DR

This paper explores defining the radial derivative of the delta distribution within spherical coordinates, introducing a new class of functionals called signumdistributions that extend standard distribution calculus.

Contribution

It introduces signumdistributions as a new class of functionals, enabling the radial derivative of the delta distribution to be defined and manipulated with familiar calculus rules.

Findings

Signumdistributions extend standard distribution calculus.

Radial derivative of delta belongs to signumdistributions.

Calculus rules for signumdistributions are straightforward and consistent.

Abstract

Possibilities for defining the radial derivative of the delta distribution $\delta(\underline{x})$ in the setting of spherical coordinates are explored. This leads to the introduction of a new class of continuous linear functionals similar to but different from the standard distributions. The radial derivative of $\delta(\underline{x})$ then belongs to that new class of so-called signumdistributions. It is shown that these signumdistributions obey easy-to-handle calculus rules which are in accordance with those for the standard distributions in $\mathbb{R}^m$.