dirac delta function

Dirac Delta Function

The Dirac Delta Function is defined to be a function {% \delta(x) %} such that

{% \delta(x) = 0 \, for \, x \neq 0 %}

{% \delta(x) = \infty \, for \, x = 0 %}

such that

{% \int_{- \infty}^{\infty} \delta(x) = 1 %}

The defintion given shows that the Dirac Delta Function is not a function in the traditional sense.

{% \int_{-\infty}^{\infty} dx f(x) \delta(x-x') = f(x') %}

One interpretation of the Dirac Delta Function is that it is the result of taking the limit of a sequence of functions.

{% \delta_n(x) = \frac{1}{\pi} \frac{sin(Nx)}{x} %}

Then, take

{% \delta(x) = lim_{N \rightarrow \infty} \frac{1}{\pi} \frac{sin(Nx)}{x} %}