Deltafunktio

Deltafunktio (Dirac delta function) is a mathematical construct used to model an idealized instantaneous impulse. In rigorous terms, it is a distribution (generalized function) rather than an ordinary function. It is defined by its action on test functions φ: ∫_{-∞}^{∞} δ(x) φ(x) dx = φ(0). Consequently, δ is not finite-valued at any point and cannot be integrated as a standard function; however, it is useful because its integral over the real line equals 1.

Key properties include the shifting and sifting behavior: ∫ δ(x - a) φ(x) dx = φ(a). Scaling holds as δ(a x) = (1/|a|) δ(x). In convolution, f * δ = f, and its Fourier transform satisfies F{δ}(ω) = 1, so the delta function is its own identity under certain transform conventions.

Representations of the delta function are typically provided as limits of families of ordinary functions that preserve area while narrowing to a point. For example, δ_ε(x) = (1/(ε√π)) e^{-(x/ε)^2} tends to δ(x) as ε → 0. Other sequences, such as Lorentzians or compactly supported bump functions, can serve similar purposes.

Applications are widespread: deltafunktio models point sources and impulse responses in physics and engineering; it appears in differential equations and Green’s functions; in signal processing and control theory it acts as the identity element under convolution; in quantum mechanics it is used to describe localized position amplitudes.

Generalizations to higher dimensions, δ^n(x), retain the same sifting property and are used in partial differential equations and related fields. The delta function was introduced by Paul Dirac in the 1930s and remains a fundamental tool across physics and mathematics.