08 - Dirac Delta

Dirac Delta

Dirac Delta function $\delta (x)$ characterised by the following properties

$$\begin{array}{rl}\delta (x)& =0\text{ for all }x\ne 0\\ {\int }_{-\infty }^{\infty }\delta (x)dx& =1\end{array}$$

Note it is also known as Delta Function

Sifting Property of Dirac Delta

For any function $\varphi (\xi )$

$${\int }_{-\infty }^{\infty }\delta (x-\xi )\varphi (\xi )d\xi =\varphi (x)$$

For $x\in (a,b)$ then

$${\int }_a^b\delta (x-\xi )\varphi (\xi )d\xi =\varphi (x)$$

Proof

$${\int }_a^b\delta (x-\xi )\varphi (\xi )d\xi ={\int }_{x-\xi }^{x+\xi }\delta (x-\xi )\varphi (\xi )d\xi$$

where $ϵ$ is an arbitrarily small positive parameter

So for smooth $\varphi (x)$ then

$${\int }_a^b\delta (x-\xi )\varphi (\xi )d\xi \sim [\varphi (x)+O(ϵ)]{\int }_{x-ϵ}^{x+ϵ}\delta (x-\xi )d\xi$$

and then as $ϵ\to 0$ we get equality to $\varphi (x)$

Approximations for Delta Function

$\delta$ can be approximated by
Sequence of increasingly narrow functions with normalised area ${\delta }_n(x)$ where

$$\begin{array}{rl}{\int }_{-\infty }^{\infty }{\delta }_n(x)dx& =1\text{ for all }n=1,2,\cdots \\ \underset{n\to \infty }{\lim }{\delta }_n(x)& =0\text{ for all }x\ne 0\end{array}$$

One example is the hat function defined by

$${\delta }_n(x)=\{\begin{array}{ll}0& \text{ for }|x|>\frac{1}{n}\\ \frac{n}{2}& \text{ for }|x|\le \frac{1}{n}\end{array}$$

Sifting Property from Approximations for Delta Function

Let $\varphi (x)$ be a smooth function
Let $\Phi (x)=\int \varphi (x)dx$ be the antiderivative

Using the hat approximating sequence then

$${\int }_{-\infty }^{\infty }{\delta }_n(x-a)\varphi (x)dx\to \varphi (a)\text{ as }n\to \infty$$

with

$${\int }_{-\infty }^{\infty }\delta (x-a)\varphi (x)dx=\varphi (a)$$

Proof

$$\begin{array}{rl}{\int }_{-\infty }^{\infty }{\partial }_n(x-a)\varphi (x)dx& ={\int }_{a-\frac{1}{n}}^{a+\frac{1}{n}}\left(\frac{n}{2}\right)\varphi (x)dx\\ & =\frac{n}{2}\left[\Phi \left(a+\frac{1}{n}\right)-\Phi \left(a-\frac{1}{n}\right)\right]\\ & =\frac{\left[\Phi \left(a+\frac{1}{n}\right)-\Phi \left(a-\frac{1}{n}\right)\right]}{\frac{2}{n}}\end{array}$$

Hence as $n\to \infty$

$${\int }_{-\infty }^{\infty }{\delta }_n(x-a)\varphi (x)dx\to {\Phi }^{\prime }(a)=\varphi (a)$$

Hence $\delta$ has the desired sifting property

$${\int }_{-\infty }^{\infty }\delta (x-a)\varphi (x)dx\equiv \varphi (a)$$

Heaviside Function

Antiderivative of delta function is the Heaviside Function

$${\int }_{-\infty }^x\delta (s)ds=H(x):=\{\begin{array}{ll}0& x<0\\ 1& x>0\end{array}$$

Note that value of $H(x)$ at $x=0$ is indeterminate, sometimes taken to be $1$ and sometimes $\frac{1}{2}$