Decomposition of multivariate function using the Heaviside step function

Springerplus. 2014 Nov 29:3:704. doi: 10.1186/2193-1801-3-704. eCollection 2014.

Abstract

Whereas the Dirac delta function introduced by P. A. M. Dirac in 1930 to develop his theory of quantum mechanics has been well studied, a not famous formula related to the delta function using the Heaviside step function in a single-variable form, also given by Dirac, has been poorly studied. Following Dirac's method, we demonstrate the decomposition of a multivariate function into a sum of integrals in which each integrand is composed of a derivative of the function and a direct product of Heaviside step functions. It is an extension of Dirac's single-variable form to that for multiple variables.

Keywords: Dirac delta function; Heaviside step function; Transform.