Definition:Dirac Delta Function

Definition
The Dirac delta function is defined as:


 * $\delta \left({t}\right) = \begin{cases}

+\infty & : t = 0 \\ 0 & : \text {otherwise} \end{cases}$

with the constraint:
 * $\displaystyle \int^{+\infty}_{-\infty} \delta \left({t}\right) \mathrm d t = \int^{0^+}_{0^-} \delta \left({t}\right) \mathrm d t = 1$

Notation
Let $c$ be a constant real number.

Some may use $\delta_c\left({t}\right)$ to denote
 * $\delta_c\left({t}\right)=\delta\left({t-c}\right)=\begin{cases}

+\infty & : t = c \\ 0 & : \text {otherwise} \end{cases}$

with the constraint:
 * $\displaystyle \int^{+\infty}_{-\infty} \delta_c \left({t}\right) \mathrm d t = \int^{c^+}_{c^-} \delta_c \left({t}\right) \mathrm d t = 1$

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

This is also called the unit pulse function.

Also see

 * Laplace Transform of Dirac Delta Function by Function:
 * $\displaystyle \mathcal L \left\{{\delta \left({t - c}\right) f \left({t}\right)}\right\} \left({s}\right) = e^{-s c} f \left({c}\right)$


 * Definition:Kronecker Delta
 * Definition:Heaviside Step Function