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DiracDelta—Wolfram Language Documentation

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• See Also
  • HeavisideTheta
  • DiracComb
  • Convolve
  • PrincipalValue
  • Limit
  • KroneckerDelta
  • DiscreteDelta
  • GreenFunction
• Related Guides
  • Generalized Functions
  • Fourier Analysis
  • Mathematical Functions
  • Integral Transforms
  • Calculus
• Tech Notes
  • Generalized Functions and Related Objects

DiracDelta[x]

represents the Dirac delta function .

DiracDelta[x₁,x₂,…]

represents the multidimensional Dirac delta function .

• DiracDelta[x] returns 0 for all real numeric x other than 0.
• DiracDelta can be used in integrals, integral transforms, and differential equations.
• Some transformations are done automatically when DiracDelta appears in a product of terms.
• DiracDelta[x₁,x₂,…] returns 0 if any of the x_i are real numeric and not 0.
• DiracDelta has attribute Orderless.
• For exact numeric quantities, DiracDelta internally uses numerical approximations to establish its result. This process can be affected by the setting of the global variable $MaxExtraPrecision.

Evaluate numerically:

DiracDelta always returns an exact 0:

Evaluate efficiently at high precision:

DiracDelta threads over lists:

As a distribution, DiracDelta does not have a specific value at 0:

Values at infinity:

Evaluate symbolically:

DiracDelta is differentiable, but its derivative does not have a special name:

Differentiate the multivariate DiracDelta:

Differentiate a composition involving DiracDelta:

Indefinite integral:

Integrate over finite domains:

Integrate over infinite domains:

Integrate expressions containing derivatives of DiracDelta:

Find classical harmonic oscillator Green function:

Solve the inhomogeneous ODE through convolution with Green's function:

Compare with the direct result from DSolve:

Define a functional derivative:

Calculate the functional derivative for an example functional:

Calculate the phase space volume of a harmonic oscillator:

Find the distribution for the third power of a normally distributed random variable:

Plot the resulting PDF:

Fundamental solution of the Klein–Gordon operator :

Visualize the fundamental solution. It is nonvanishing only in the forward light cone:

A cusp‐containing solution of the Camassa–Holm equation:

Higher derivatives will contain DiracDelta:

Plot the solution and its derivative:

Differentiate and integrate a piecewise defined function in a lossless manner:

Differentiating and integrating recovers the original function:

Using Piecewise does not recover the original function:

Solve a classical second‐order initial value problem:

Incorporate the initial values in the right‐hand side through derivatives of DiracDelta:

Only HeavisideTheta gives DiracDelta after differentiation:

This also holds for the multivariate case:

DiracDelta[0] is not an "infinite" quantity:

DiracDelta can stay unevaluated for numeric arguments:

Products of distributions with coinciding singular support cannot be defined:

DiracDelta cannot be uniquely defined with complex arguments:

Numerical routines will typically miss the contributions from measures at single points:

Limit does not produce DiracDelta as a limit of smooth functions:

Integrate never gives DiracDelta as an integral of smooth functions:

FourierTransform can give DiracDelta:

Calculate the moments of a Gaussian bell curve:

Do it using the dual Taylor expansion expressed in derivatives of DiracDelta:

The two sequences of moments are identical:

Introduced in 1999 (4.0)

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