Definition:Dirac Delta Function

Also defined as
Let $c$ be a constant real number.

The notation $\map {\delta_c} t$ is often used to denote:
 * $\map {\delta_c} t := \map \delta {t - c} := \begin{cases}

0 & : x < c - \epsilon \\ \dfrac 1 {2 \epsilon} & : c - \epsilon \le x \le c + \epsilon \\ 0 & : x > c + \epsilon \end{cases}$

Also see
The Dirac delta function is also defined by the following limits:

The Dirac delta function has the following fundamental property:
 * $\ds \map f s = \int_{-\infty}^\infty \map f x \map \delta {x - s} \rd x$

If $\epsilon > 0$, then:
 * $\ds \map f s = \int_{s - \epsilon}^{s + \epsilon} \map f x \map \delta {x - s} \rd x$


 * Laplace Transform of Dirac Delta Function
 * Laplace Transform of Shifted Dirac Delta Function
 * Laplace Transform of Dirac Delta Function by Function:
 * $\laptrans {\map \delta {t - c} \map f t} = e^{-s c} \map f c$


 * Definition:Kronecker Delta
 * Definition:Heaviside Step Function