# Dirac Delta Function ![[DiracDeltaFunction.svg]] *The Dirac delta function is typically depicted as an arrow with length proportional to its multiplicative constant* > [!NOTE] Dirac Delta Function > The *Dirac delta function*, $\delta(x)$, also known as the *unit impulse function*, is zero for all values except at $x=0$, where it is undefined. > > That is, > $\delta(x)=\begin{cases}0,&x\neq0\\\infty,&x=0\end{cases}$ > such that > $\int_{-\infty}^{\infty}\delta(x)\;dx=1$ An important application of the Dirac delta function its *sampling* or *sifting property*. The [[integral]] of any function multiplied by the Dirac delta function shifted to $x_{0}$ will give the value of the function at $x_{0}$. $\int_{-\infty}^{\infty}f(x)\delta(x-x_{0})\;dx=f(x_{0})$