DiracDelta

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

Specialized values

Values at fixed points

General characteristics

Domain and analyticity

δ(x) is a non-analytical function; it is generalized function defined for x∈ℝ.

Symmetries and periodicities

Parity

δ(x) is an even generalized function.

Periodicity

No periodicity

Series representations

Exponential Fourier series

Integral representations

On the real axis

Of the direct function

Limit representations

Transformations

Transformations and argument simplifications

Identities

Functional identities

Differentiation

Symbolic differentiation

In a distributional sense, for x∈ℝ :

Integration

Indefinite integration

Involving only one direct function

Definite integration

For the direct function itself

Involving the direct function

In the following formulas a∈ℝ .

Integral transforms

Fourier exp transforms

Inverse Fourier exp transforms

Fourier cos transforms

Fourier sin transforms

Laplace transforms

Summation

Infinite summation

Representations through more general functions

Through Meijer G

Classical cases for the direct function itself

Representations through equivalent functions

Theorems

Sokhotskii's formulas

(lim_(ε->0))(1/(x±i ε))==∓i π δ(x)+𝒫((1/x))

Green's function of one linear differential operator

Let  (Subscript[(OverHat[L]), x]) be a linear differential operator. The fundamental solution G(x-ξ) (also called Green's function) of (Subscript[(OverHat[L]), x]) fulfills the equation (Subscript[(OverHat[L]), x])G(x-ξ)==δ(x-ξ). Then the solution to the equation (Subscript[(OverHat[L]), x])y(x)==f(x) can be represented in the form y(x)==∫f(ξ)G(x-ξ)dξ.

The left eigenstates of the Fröbenius-Perron operator

The functional  (Subscript[ψ, n])(x)==(((-1))^(n-1))((δ^((n-1)))(x-1)-(δ^(n-1))(x))/;n∈(SuperPlus[ℕ]) are the left eigenstates of the Fröbenius-Perron operator for the r-adic map (Subscript[x, n+1])==r (Subscript[x, n]) mod 1.

Poincaré-Bertrand theorem

(Underscript[(lim_((Subscript[ε, 1])->0)), (Subscript[ε, 2])->0])(1/((Subscript[x, 1])-i (Subscript[s, 1])ε))(1/((Subscript[x, 2])-i (Subscript[s, 2])ε))==(𝒫((1/(Subscript[x, 1])))+i π (Subscript[s, 1])δ((Subscript[x, 1])))(𝒫((1/(Subscript[x, 2])))+i π (Subscript[s, 2])δ((Subscript[x, 2])))+(π^2)δ((Subscript[x, 1]))δ((Subscript[x, 2]))/;(Subscript[s, 1]),(Subscript[s, 2])==±1

The generalized solutions of the linear differential operators with singular coefficients

Linear differential operators with singular coefficients can have generalized solutions. For the hypergeometric differential equation x(1-x)  y''(x)+  (γ-(α+β+1)x)y'(x)-α β y(x)==0 for α, β, γ∈(SuperPlus[ℕ]), α≥γ>β it can be presented in the form

y(x)=(Subscript[c, 1])(∑_(k=0)^(γ-β-1))(((((-1))^k)(Subscript[(β-γ-1), k]))/(k!(Subscript[(β-α-1), k])))(δ^((β+k-1)))(x)+(Subscript[c, 2])(∑_(k=0)^∞)(((((-1))^k)(Subscript[(α-γ-1), k]))/(k!(Subscript[(α-β-1), k])))(δ^((β+k-1)))(x)

The functional derivative of a function

The functional derivative of a function f(x) is ((δ f(x))/(δ f(y)))==δ(x-y).

History

– O. Heaviside (1893–1895)
– G. Kirchhoff (1891)
– P. A. M. Dirac (1926)
– L. Schwartz (1945).

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