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.

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