The Cluster's Dirac Delta Function

A page about the Dirac delta function — the distribution that is zero everywhere except at one point and integrates to one.

The Dirac delta function

The Dirac delta function is defined by its property under an integral: integral_{-infinity}^{infinity} delta(x) f(x) dx = f(0) for any test function f(x). The delta function is zero for x != 0 and "infinite" at x = 0, but the infinity is such that the integral is 1. More precisely, delta(x) is a distribution (generalized function), not a function in the classical sense.

In the cluster, the edit Dirac delta function is defined by an edit integral property.

The representation as a limit

The delta function can be represented as the limit of various sequences:

1. Gaussian: delta(x) = lim_{sigma->0} (1 / (sqrt(2 pi) sigma)) exp(-x^2 / (2 sigma^2))
2. Rectangle: delta(x) = lim_{epsilon->0} (1 / (2 epsilon)) for |x| < epsilon, 0 otherwise
3. Sinc: delta(x) = lim_{L->infinity} sin(Lx) / (pi x)
4. Lorentzian: delta(x) = lim_{epsilon->0} (epsilon / pi) / (x^2 + epsilon^2)

In the cluster, the edit delta function is a limit of edit sequences.

The Fourier transform

The Fourier transform of the delta function: integral delta(x) exp(-ikx) dx = 1 The inverse transform: (1 / 2pi) integral exp(ikx) dk = delta(x) The delta function appears in Fourier series completeness relations and in momentum-space representations.

In the cluster, the edit Fourier transform gives an edit completeness relation.

Properties

• Scaling: delta(ax) = (1 / |a|) delta(x)
• Derivative: integral delta'(x) f(x) dx = -f'(0)
• Multiplication: delta(x) f(x) = f(0) delta(x)
• Composition: delta(g(x)) = sum_i delta(x - x_i) / |g'(x_i)| where x_i are the zeros of g(x)
• The sifting property: integral delta(x - a) f(x) dx = f(a)

In the cluster, the edit properties give an edit sifting relation.

Applications

• Point sources in differential equations (Poisson's equation: nabla^2 V = -rho / epsilon_0)
• Impulse forces in mechanics (F(t) = I delta(t - t_0))
• Initial conditions in PDEs
• Scattering theory (potential V(x) = V_0 delta(x))
• Probability (discrete random variables)

In the cluster, edit applications include:

• edit Point sources
• edit Impulse forces
• edit Initial conditions
• edit Scattering theory
• edit Probability

This function

This page is about the Dirac delta function. integral delta(x) f(x) dx = f(0). delta(ax) = (1/|a|) delta(x). Fourier transform = 1. The function is real.