Dirac delta function

"Delta function" redirects here. For other uses, see Delta function (disambiguation).

Schematic representation of the Dirac delta function by a line surmounted by an arrow. The height of the arrow is usually meant to specify the value of any multiplicative constant, which will give the area under the function. The other convention is to write the area next to the arrowhead.

In mathematical analysis, the Dirac delta function (or ${\boldsymbol {\delta }}$ distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, where it is infinite, and whose integral over the entire real line is equal to one. Thus it can be represented heuristically as$$\delta (x)={\begin{cases}0,&x\neq 0\\{\infty },&x=0\end{cases}}$$such that$$\int _{-\infty }^{\infty }\delta (x)\,dx=1.$$

Since no function has this property, modelling the delta "function" rigorously involves the use of limits or, as is common in mathematics, measure theory and the theory of distributions.

The delta function is named after physicist Paul Dirac, and has been applied routinely in physics and engineering to model point masses and concentrated loads. It is called the delta function because it is a continuous analogue of the Kronecker delta function. The mathematical rigor of the delta function was disputed until Laurent Schwartz developed the theory of distributions, where it is defined as a linear form acting on functions.