Dirac Delta Function - Distributional Derivatives

Distributional Derivatives

The distributional derivative of the Dirac delta distribution is the distribution δ′ defined on compactly supported smooth test functions φ by

The first equality here is a kind of integration by parts, for if δ were a true function then

The k-th derivative of δ is defined similarly as the distribution given on test functions by

In particular δ is an infinitely differentiable distribution.

The first derivative of the delta function is the distributional limit of the difference quotients:

More properly, one has

where τ_h is the translation operator, defined on functions by τ_hφ(x) = φ(x+h), and on a distribution S by

In the theory of electromagnetism, the first derivative of the delta function represents a point magnetic dipole situated at the origin. Accordingly, it is referred to as a dipole or the doublet function.

The derivative of the delta function satisfies a number of basic properties, including:

Furthermore, the convolution of δ' with a compactly supported smooth function ƒ is

which follows from the properties of the distributional derivative of a convolution.