deltafunction

The Dirac delta function, denoted δ(x), is a distribution rather than an ordinary function. It is defined by its action on smooth test functions φ: ∫_{−∞}^{∞} δ(x) φ(x) dx = φ(0). Informally, δ(x) is zero for x ≠ 0 and integrates to one, concentrating all mass at the origin.

One of its central properties is the sifting (or sampling) property: ∫ δ(x − a) φ(x) dx = φ(a). This extends to higher dimensions as δ^n(x) with ∫ δ^n(x) φ(x) d^n x = φ(0). Scaling and shifts follow the rules δ(a x) = δ(x)/|a| and δ(x − a) behaves similarly under integration with a test function.

Delta functions can be represented as limits of ordinary functions. Common approximations include δ_ε(x) = (1/(ε√π)) e^{−(x^2/ε^2)} and δ_ε(x) = (1/π) (ε/(x^2 + ε^2)), with δ_ε → δ in the sense of distributions as ε → 0. These approximations satisfy ∫ δ_ε(x) φ(x) dx → φ(0).

In the Fourier domain, the delta function is its own universal sampler: its transform is constant, δ̂(k) = 1 (up to conventional factors depending on Fourier normalization). Consequently, δ acts as the identity under convolution: f * δ = f.

Derivatives of the delta function are defined through distributional action: ⟨δ', φ⟩ = −⟨δ, φ'⟩ = −φ′(0). Thus ∫ δ'(x) φ(x) dx = −φ′(0). Generalizations to higher dimensions and to more general arguments follow analogous distributional rules.

Applications span physics and engineering, including impulse responses, Green’s functions, signal processing, and quantum mechanics. The delta function is a mathematical idealization used to model instantaneous, localized sources. It is a distribution, not a conventional function, and its precise meaning is via its action on test functions.