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


which has the integral


for all t>0 .

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


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


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


is useful in physics for modelling systems of point charges.