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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