Dirac delta function

File:Dirac distribution PDF.svg

Schematic representation of the Dirac delta function by a line surmounted by an arrow. The height of the arrow is usually used 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.

File:Dirac function approximation.gif

The Dirac delta function as the limit (in the sense of distributions) of the sequence of zero-centered normal distributions [math]\displaystyle{ \delta_a(x) = \frac{1}{a \sqrt{\pi}} \mathrm{e}^{-x^2/a^2} }[/math] as [math]\displaystyle{ a \rightarrow 0 }[/math].

The Dirac delta function is a made-up concept by mathematician Paul Dirac. It is a really pointy and skinny function that pokes out a point along a wave. The delta function is used a lot in sampling theory where its pointiness is useful for getting clean samples.

The integral of the Dirac Delta Function is the Heaviside Function.