the Dirac delta distribution
/ dee-RAHK /
The Dirac delta is the precise idea of an instantaneous point: a unit of something packed entirely at a single location, with nothing anywhere else. A point charge, a hammer blow at one instant, a unit mass at the origin — physicists long described these by a 'function' that is infinite at one point, zero everywhere else, and yet has total integral one. No such function exists. The delta makes the idea legal by recasting it as a distribution.
As a distribution, the delta is defined not by values but by what it does to a probe: pairing the delta with any test function phi simply returns the probe's value at the origin, (delta, phi) = phi(0). That single rule is the whole object. A shifted delta concentrated at a point a does the same at that point, (delta-a, phi) = phi(a). You can picture the delta as the limit of taller and thinner bumps — each a real function with integral one, getting narrower and taller — so that the integral of bump times phi homes in on phi(0). The delta is the idealized endpoint of that squeezing, captured exactly rather than approximately.
The delta is everywhere in PDEs because it is the perfect point source. The fundamental solution of an operator is, by definition, its response to a delta source; the heat kernel is the temperature from a unit of heat released at one point and instant; a Green's function is a delta response shaped to fit boundary conditions. Under the Fourier transform the delta becomes the constant 1, which is why a point in space corresponds to all frequencies equally — the cleanest example of the transform of a distribution.
Take bumps b_n(x) = n for |x| < 1/(2n) and 0 elsewhere — each a thin tall box of area 1. As n grows, the integral of b_n times phi converges to phi(0) for every smooth probe phi. The delta is the exact limit of this process: (delta, phi) = phi(0), no approximation needed.
The delta is the captured limit of narrowing unit-area bumps — its reading is always the probe's value at the spike.
The delta is not a function and has no value at 0; writing 'delta(0) = infinity' is a fiction. Likewise the integral of delta times phi is shorthand for the pairing phi(0), not a real Riemann or Lebesgue integral of a function.