A mathematical interpretation of Dirac δ function

Abstract

The two conditions (see[1] p. 58)

$$\delta (x) = 0,for x \ne 0$$

(1.1)

$$\int_{ - \infty }^{ + \infty } {\delta (x)dx = 1} $$

(1.2)

of the Dirac δ function are inconsistent in standard analysis.

In this paper, the author began by studying the integral of the functions on the nucleon α(0), and then, making use of the point function in infinitesimal analysis to define the Dirac δ function δ(x) so that it satisfies the condition (1.2) and δ(x)=0, forx∈R and x≠0

Some various examples of Dirac δ functions have been presented and some properties of the δ function have been derived.