# Thread: Dirac delta function

1. ## Dirac delta function

Could someone explain to me why $\int_{-\infty}^\infty f(x)\delta(x)\, dx = f(0)$.

2. Originally Posted by chiph588@
Could someone explain to me why $\int_{-\infty}^\infty f(x)\delta(x)\, dx = f(0)$.
Which definition of the Dirac delta function are you using? The one with limits?

3. Originally Posted by Drexel28
Which definition of the Dirac delta function are you using? The one with limits?
$\delta(x) = \begin{cases} +\infty, & x = 0 \\ 0, & x \ne 0 \end{cases}$

s.t. $\int_{-\infty}^\infty \delta(x) \, dx = 1$.

I think I have a solution, but I'm afraid the way $\delta$ is defined makes my argument invalid.

4. Originally Posted by chiph588@
$\delta(x) = \begin{cases} +\infty, & x = 0 \\ 0, & x \ne 0 \end{cases}$

s.t. $\int_{-\infty}^\infty \delta(x) \, dx = 1$.

I think I have a solution, but I'm afraid the way $\delta$ is defined makes my argument invalid.
Well, to be honest with you I am no expert on this field. But, I was under the impression that the integral you quoted is an abuse of notation considering the way $\delta$ is defined makes it impossible for $\int_{-\infty}^{\infty}f(x)\delta(x)dx$ to be an actual Riemann or Lebesgue integral.

5. Well I suppose $\delta(x) = \lim_{a\to 0}\frac{1}{a \sqrt{\pi}} \mathrm{e}^{-x^2/a^2}$.