$\delta (t)=\{\begin{array}{c}0\\ \infty \\ 0\end{array}\begin{array}{c}\text{for}t<0\\ \text{for}t=0\\ \text{for}t>0\end{array}\text{with}{\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}\delta (t)}\mathrm{d}t=1$

SEE: Figure 4a) and IEC 60027-6.

Note 1 to entry: A distribution assigns a number to any function f(t), sufficiently smooth for t = t₀ [see CEI 60050-103:2009, 103-03-05].

The Dirac impulse does this according to

${\displaystyle f\left(t_0\right)={\displaystyle \underset{-\infty }{\overset{\infty }{\int }}\delta }\left(t-t_0\right)f\left(t\right)\text{d}t}$.

Note 2 to entry: Any shape with area 1 may be used for the definition of δ(t), e.g. a rectangular pulse with width τ and height τ^–1, or a triangular pulse, as shown in Figure 4a), as well as a Gaussian function

${\displaystyle \frac{1}{\tau \cdot \sqrt{\pi }}\cdot e^{-\frac{t^2}{{\tau }^2}}}$.

Note 3 to entry: Any of the shapes mentioned in Note 2 to entry with τ much smaller than the smallest time constant at work in the system under consideration may be used for a technical approximation of the Dirac impulse.

Note 4 to entry: In control technology the Dirac function is mainly important for the definition of impulses and exclusively used as a function of time. Therefore the term Dirac impulse is used and the definition is adapted accordingly.