FANDOM


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.

Ad blocker interference detected!


Wikia is a free-to-use site that makes money from advertising. We have a modified experience for viewers using ad blockers

Wikia is not accessible if you’ve made further modifications. Remove the custom ad blocker rule(s) and the page will load as expected.