Dirac Delta Function

The Dirac Delta Function could be applied to simplify the differential equation. There are three main properties of Dirac Delta Function.

$$\delta (x-x’) =\lim_{\tau\to0}\delta (x-x’)$$

such that,

$$ \delta (x-x’) = \begin{cases} \infty & x= x’ \ 0 & x\neq x’ \end{cases} $$

$$\int_{-\infty}^{\infty} \delta (x-x’)\ dx =1$$

Three Properties:

  • Property 1:

$$\delta(x-x’)=0 \quad \quad ,x\neq x’ $$

  • Property 2:

$$ \int_{x’-\epsilon}^{x’+\epsilon} \delta (x-x’)dx =1\quad \quad ,\epsilon >0 $$

  • Property 3:

$$\int_{x’-\epsilon}^{x’+\epsilon} f(x)\ \delta (x-x’)dx = f(x’)$$

At x=x’ the Dirac Delta function is sometimes thought of has having an “infinite” value. So, the Dirac Delta function is a function that is zero everywhere except one point and at that point it can be thought of as either undefined or as having an “infinite” value.