A - Delta Function

Summary

As a function of the variable x the Dirac delta function, or simply the delta function, denoted by δ(x), is defined as follows:

The condition on the ei's means that the limit of integration in (A.1) should include origin. The ϵ1, ϵ2 in (A.1) can have any positive value and may even be infinitesimally close to zero. It then follows that for f (x) continuous at the origin (a, b, ϵ > 0, ϵ ½ 0),

The second equation above follows by the use of the first part of (A.1), and the third from the fact that, due to continuity of f (x) at x = 0, the value of f (x) for x 2 (−ϵ, ϵ) can be taken to be f (0) as ϵ → 0. In general

The derivative of the delta function is defined as follows:

The higher derivatives can be defined in similar manner.

The defining relations of the delta function and its derivative may be written symbolically as follows:

where prime denotes the derivative with respect to x.

We may similarly define the three-dimensional delta function δ⁽³⁾(r−r), where r, rÌ are the position vectors, by the equation

where the integral is over the volume containing r. Let us express δ⁽³⁾(rÌ − r) in terms of one-dimensional delta functions. We will see that the said expression depends on the coordinate system.