Definition:Dirac Delta Function

Definition
Let $\epsilon \in \R_{>0}$ be a (strictly) positive real number.

Consider the real function $F_\epsilon: \R \to \R$ defined as:


 * $\map {F_\epsilon} x = \begin{cases}

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

The Dirac delta function is defined as:


 * $\map \delta x = \displaystyle \lim_{\epsilon \mathop \to 0} \map {F_\epsilon} x$

However, note that while $\map \delta x$ is usually so referred to as a function and treated as a function, it really is not actually a function at all.

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

Also known as
Less commonly it is rendered as Dirac's delta function.

This is also called the unit pulse function or unit impulse function.

Some sources refer to $\map \delta x$ just as the impulse function.

Some, acknowledging the fact that it is not actually a function as such, refer to it as the unit impulse.

Also see

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


 * Definition:Kronecker Delta
 * Definition:Heaviside Step Function