Dirac delta function
The Dirac delta, introduced by Paul Dirac, can be thought of as a function δ(x) that has the value of infinity for x = 0, the value zero elsewhere, and a total integral of one. The graph of the delta function can be thought of as following the whole x-axis and the positive y-axis.
The Dirac delta is very useful as an approximation for tall narrow spike functions. It is the same type of abstraction as a point charge, point mass or electron point. For example, in calculating the dynamics of a baseball being hit by a bat, approximating the force of the bat hitting the baseball by a delta function is a helpful trick. In doing so, one not only simplifies the equations, but one also is able to calculate the motion of the baseball by only considering the total impulse of the bat against the ball rather than requiring knowledge of the details of how the bat transferred energy to the ball.
The Dirac delta is often introduced with the property:
However, there is no function δ(x) with this property. Technically speaking, the Dirac delta is not a function but a distribution which is a mathematical expression that is well defined only when integrated. As a distribution, the Dirac delta is defined by
The Dirac delta distribution is the derivative of the Heaviside step function,
The derivative of the Dirac delta is the distribution δ' defined by
Interestingly, the delta function is also given by the identity:
Dirac delta function is a distribution whose image is homeomorphic to the function and satisfies the integral equation as follows: