Dirac delta function

The Dirac delta function, introduced by Paul Adrien Maurice Dirac (1902-1984), can be thought of as a function $\delta (x)$ that has the value of infinity for $x=0$, the value zero elsewhere, and by convention a total integral of one. It 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 function is often introduced with the property:

$\int _{-\infty }^{\infty }f(x)\,\delta (x)\,dx=f(0)$

valid for any infinitely often differentiable function $f$.

However, there is no function $\delta (x)$ 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

$\delta (\phi )=\phi (0)$

for every test function $\phi$. It is a distribution with compact support (the support being {0}).

The Dirac delta distribution is the derivative of the Heaviside unit step function,

$H(x)=\left\{{\begin{matrix}0:x<0\\1:x\geq 0\end{matrix}}\right.$

if one defines the term "derivative" in the proper, distribution-theoretic sense. (Using the ordinary definition of derivative from calculus, $H(x)$ is not differentiable for $x=0$.)

The Fourier transform of the Dirac delta is the constant function 1, and the convolution of $\delta$ with any distribution $S$ yields $S$.

The derivative of the Dirac delta is the distribution $\delta '$ defined by

$\delta '(\phi )=-\phi '(0)$

for every test function $\phi$. The $n$-th derivative δ⁽ⁿ⁾ is given by

$\delta ^{(n)}(\phi )=(-1)^{n}\phi ^{(n)}(0)$

The derivatives of the Dirac delta are important because they appear in the Fourier transforms of polynomials.

Interestingly, the delta function is also given by the identity:

$\delta (x)=\lim _{\epsilon \to 0}{\epsilon \over \epsilon ^{2}+x^{2}}$

From this definition it is trivial to derive two important properties:

1. $\delta (x)=0{\mbox{ for }}x\neq 0$
2. $\delta (0)=\infty$

The graph of the delta function can be thought of as following the whole x-axis and the positive y-axis. If you integrate the delta function between ANY limits a and b, then the integral is:

0   if a,b > 0 or a,b < 0

1   if a < 0 < b

0.5   if a = 0 or b = 0 (not both) (the truth and meaning of this depending on exact definitions)