physica.
Mathematical Physicsundergraduategraduate

Dirac Delta Function

Also known as: Delta function · Unit impulse · Dirac distribution

The delta function is an idealized spike with zero width and unit area. Its defining job is the sifting property: integrated against any function, it samples that function at a single point.

∫−∞∞f(x) δ(x−a) dx=f(a)\int_{-\infty}^{\infty} f(x)\,\delta(x-a)\,dx = f(a)
Live simulation
warming up the physics…

A normalized Gaussian (the nascent delta) narrows and grows as the width slider shrinks, its area pinned at 1. A smooth test function f(x) (gray) is shown; a marker reads off the sampled value f(a) that the spike selects — the sifting property made visible.

Equivalent forms

δ(x)=0 (x≠0),∫−∞∞δ(x) dx=1\delta(x) = 0\ (x\neq 0),\quad \int_{-\infty}^{\infty}\delta(x)\,dx = 1
δ(x)=lim⁡ε→012πεe−x2/2ε2\delta(x) = \lim_{\varepsilon\to 0}\frac{1}{\sqrt{2\pi}\varepsilon}e^{-x^2/2\varepsilon^2}
δ(x)=12π∫−∞∞eikx dk\delta(x) = \frac{1}{2\pi}\int_{-\infty}^{\infty} e^{ikx}\,dk
It is the identity element of convolution and the continuous analogue of the Kronecker delta — the bridge between discrete sums and continuous integrals.