Dirac delta ($\delta(t)$)

A generalized function, the limit of ever-narrower unit-area pulses, defined by what it does inside an integral.

The Dirac delta, $\delta(t)$ , is not a function in the ordinary sense -- it is a generalized function (distribution), the limit of a family of narrowing, unit-area pulses concentrated entirely at $t=0$ . It is defined operationally by its sifting property rather than by a pointwise formula.

Examples

$\delta(t-3)$ models an idealized instantaneous impulse arriving at time $t=3$ .