DiracDelta

DiracDelta, usually written δ(x) or δ, refers to the Dirac delta distribution, an idealized object used to model an instantaneous impulse or a point source. It is not a function in the traditional sense; rather, it is defined by how it acts on test functions in the theory of distributions. Informally, δ(x) is zero for all x ≠ 0 and its total integral over the real line is one.

The defining property is the sifting (or sampling) integral: for any smooth, rapidly decaying function f, ∫_{-∞}^{∞} f(x) δ(x − a) dx = f(a). This expresses that δ picks out the value of f at the point a. The delta is translation-invariant in the sense that δ(x − a) represents a unit impulse at position a. It acts as the identity element under convolution: f * δ = f.

Key identities include the scaling relation δ(a x) = (1/|a|) δ(x) for a ≠ 0, and the derivative δ'(x), defined by ∫ f(x) δ'(x) dx = − f'(0) for smooth f. Accordingly, ∫ f(x) δ'(x − a) dx = − f'(a). The delta is the distributional derivative of the Heaviside step function H, since δ = dH/dx in the distribution sense.

In the Fourier transform framework, the transform of δ is a constant function (depending on convention): typically, the transform is 1, indicating that δ contains all frequency components equally. In practice, δ is often approached as a limit of families of functions, such as Gaussian functions with variance tending to zero, or other approximate identities, which converge to δ in the distribution sense.

Applications span physics and engineering, including impulse responses, Green’s functions, and signal processing, where δ models instantaneous input, point charges, or localized sources.