# Dirac delta function

The Dirac delta function as the limit (in the sense of distributions) of the sequence of zero-centered normal distributions ${\displaystyle \delta _{a}(x)={\frac {1}{a{\sqrt {\pi }}}}\mathrm {e} ^{-x^{2}/a^{2}}}$ as ${\displaystyle a\rightarrow 0}$.
The Dirac delta function, often written as ${\displaystyle \delta (x)}$, 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. Loosely speaking, it has the value of zero everywhere except at ${\displaystyle x=0}$, in such a way that the area between the function and the x-axis adds up to 1.[1] The delta function is often used in sampling theory, where its pointiness is useful for getting clean samples.