Dirac delta function

Generalized function whose value is zero everywhere except at zero

In mathematical analysis, the Dirac delta function (or δ {\displaystyle {\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 δ ( x) = { 0 , x ≠ 0 ∞ , x = 0 {\displaystyle \delta (x)={\begin{cases}0,&x\neq 0\\{\infty },&x=0\end{cases}}} such that ∫ − ∞ ∞ δ ( x) d x = 1. {\displaystyle \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.

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