Dirac delta function

The Dirac delta function, often represented as $\delta(x)$ , is a mathematical object (not technically a function) that is defined as

$\delta(x)=\begin{cases}\infty&x=0\\0&x\ne0\end{cases}$

which has the integral

$\int\limits_{-t}^t\delta(x)dx=1$

for all $t > 0$ .

It is also the derivative of the Heaviside function, which can be written as

$u_c(t)=\int\limits_{-\infty}^t\delta(s-c)ds$

It can be defined as the limit of a normal distribution as it gets steeper and steeper, or the limit as $t\to0$ of the function

$\delta(x)=\begin{cases}\tfrac{1}{2t}&|x|\le t\\0&|x|>t\end{cases}$

It has the Laplace transform

$\mathcal{L}\{u_c(t)\}=e^{-cs}$

for $c > 0$ .

The Dirac delta function is often used in differential equations to approximate physical actions that take place over very short time intervals, such as a bat striking a ball. The 3D Dirac delta function, defined as

$\delta(\mathbf{r}-\mathbf{r}_0)=\begin{cases}\infty&\mathbf{r}=\mathbf{r}_0\\ 0&\mathbf{r}\ne\mathbf{r}_0\end{cases}$

is useful in physics for modelling systems of point charges.