Dirac delta function

The Dirac delta function, introduced by Paul Dirac, can be informally 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:

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

valid for any continuous function f.

However, there is no function δ(x) with this property. Technically speaking, the Dirac delta is not a function but a distribution — a mathematical expression that is well defined only when integrated. As a distribution, the Dirac delta is defined by

${\displaystyle \delta [\phi ]=\phi (0)}$

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

It is also convenient to think of the delta as a functional, defined by

${\displaystyle \delta [f(x)]=f(0)}$

Or literally, for every test function f(x) it return f 's value in x=0.

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

${\displaystyle H(x)=\left\{{\begin{matrix}0:x<0\\{\frac {1}{2}}:x=0\\1:x>0\end{matrix}}\right.}$

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 δ with any distribution S yields S.

The derivative of the Dirac delta is the distribution δ' defined by

${\displaystyle \delta '[\phi ]=-\phi '(0)}$

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

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

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

A helpful identity is

${\displaystyle \delta (g(x))=\sum _{i}{\frac {\delta (x-x_{i})}{|(g'(x_{i})|}}}$

where ${\displaystyle x_{i}}$ are the roots of g(x). In the integral form it is equivalent to

${\displaystyle \int _{-\infty }^{\infty }f(x)\,\delta (g(x))\,dx=\sum _{i}({\frac {f(x_{i})}{|(g'(x_{i})|}})}$

The Dirac delta function ${\displaystyle \delta :\mathbb {R} \ni \xi \longrightarrow \delta (\xi )\in \delta (\mathbb {R} )\subset \mathbb {R} }$ is a distribution ${\displaystyle \delta (\xi )}$ whose indefinite integral is the function

${\displaystyle h:\mathbb {R} \ni \xi \longrightarrow {\frac {1+{\rm {sgn}}\xi }{2}}\in \mathbb {R} ,}$

usually called the Heaviside function. That is, it satisfies the integral equation

${\displaystyle \int _{-\infty }^{x}\delta (t)dt=h(x)}$

for all real numbers x.

The delta function can be viewed as the limit of a sequence of functions

${\displaystyle \delta (x)=\lim _{a\to 0}\delta _{a}(x),}$

where ${\displaystyle \delta _{a}(x)}$ is sometimes called a nascent delta function. This may be useful in specific applications; to put it another way, one justification for the delta-function notation is that it doesn't presuppose which limiting process will be used. On the other hand if the term limit is used too loosely, nonsense may result here, as in any branch of mathematical analysis. The term approximate identity has a particular meaning in harmonic analysis, in relation to a limiting sequence to an identity element for the convolution operation (on groups more general than the real numbers, e.g. the unit circle). There the condition is made that the limiting sequence should be of positive functions.

Some nascent delta functions are:

${\displaystyle \delta _{a}(x)={\frac {1}{\pi }}{a \over a^{2}+x^{2}}}$

${\displaystyle \delta _{a}(x)=\left\{{\begin{matrix}{\frac {1}{2a}}:-a<x<a\\0:\mathrm {otherwise} \end{matrix}}\right.}$

${\displaystyle \delta _{a}(x)={\frac {1}{a{\sqrt {\pi }}}}\mathrm {e} ^{-x^{2}/a^{2}}}$

${\displaystyle \delta _{a}(x)=\partial _{x}{\frac {1}{1+\mathrm {e} ^{-x/a}}}=-\partial _{x}{\frac {1}{1+\mathrm {e} ^{x/a}}}}$

${\displaystyle \delta _{a}(x)={\frac {a}{\pi x^{2}}}\sin ^{2}\left({\frac {x}{a}}\right)}$

${\displaystyle \delta _{a}(x)={\frac {1}{\pi x}}\sin \left({\frac {x}{a}}\right)={\frac {1}{2\pi }}\int _{-1/a}^{1/a}\cos(kx)\;dk}$

${\displaystyle \delta _{a}(x)={\frac {1}{\pi }}{\frac {1}{x^{2}+a^{2}}}}$

${\displaystyle \delta _{a}(x)={\frac {1}{2\pi }}\int _{-\infty }^{\infty }\mathrm {e} ^{\mathrm {i} kx-a|k|}\;dk}$

• Kronecker delta
• Dirac comb