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

which has the integral

for all .

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 of the function

It has the Laplace transform

for .

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.