Dirac measure / Dirac delta¶
It is an infinitely narrow gaussian centered at an point. Mathematically:
\[\begin{split}
\delta_x (A) = \begin{cases}
1 & \text{ if } x \in A \\
0 & \text { if } x \notin A
\end{cases}
\end{split}\]
Such that:
\[ \int_{-\infty}^{\infty} \delta(x)dx = 1 \]
Usage¶
It can be used to select a single term from an integral:
\[ \int_{-\infty}^{\infty} f(x)\delta(x - \mu)dx = f(\mu)\]