DiracDelta

DiracDelta[x]

represents the Dirac delta function .

DiracDelta[x1,x2,…]

represents the multidimensional Dirac delta function .

Details

  • 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 xi 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.

Examples

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Basic Examples  (3)

DiracDelta vanishes for nonzero arguments:

DiracDelta stays unevaluated for :

Plot over a subset of the reals:

Use DiracDelta in an integral:

Scope  (22)

Numerical Evaluation  (4)

Evaluate numerically:

DiracDelta always returns an exact 0:

Evaluate efficiently at high precision:

DiracDelta threads over lists:

Specific Values  (3)

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

Values at infinity:

Evaluate symbolically:

Function Properties  (4)

Function domain of DiracDelta:

It is restricted to real arguments:

DiracDelta is an even function:

DiracDelta has unit area despite being zero everywhere except the origin:

TraditionalForm formatting:

Differentiation  (3)

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

Differentiate the multivariate DiracDelta:

Differentiate a composition involving DiracDelta:

Integration  (4)

Indefinite integral:

Integrate over finite domains:

Integrate over infinite domains:

Integrate expressions containing derivatives of DiracDelta:

Integral Transforms  (4)

Find the FourierTransform of DiracDelta:

Find the FourierTransform of a shifted DiracDelta:

Find the LaplaceTransform of DiracDelta:

Find the MellinTransform of DiracDelta:

DiracDelta is the identity element of Convolve:

Applications  (8)

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:

Properties & Relations  (4)

Expand DiracDelta into DiracDelta with linear arguments:

Simplify expressions containing DiracDelta:

Fourier transforms:

Laplace transforms:

Possible Issues  (8)

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:

Neat Examples  (1)

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:

Wolfram Research (1999), DiracDelta, Wolfram Language function, https://reference.wolfram.com/language/ref/DiracDelta.html.

Text

Wolfram Research (1999), DiracDelta, Wolfram Language function, https://reference.wolfram.com/language/ref/DiracDelta.html.

CMS

Wolfram Language. 1999. "DiracDelta." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/DiracDelta.html.

APA

Wolfram Language. (1999). DiracDelta. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/DiracDelta.html

BibTeX

@misc{reference.wolfram_2024_diracdelta, author="Wolfram Research", title="{DiracDelta}", year="1999", howpublished="\url{https://reference.wolfram.com/language/ref/DiracDelta.html}", note=[Accessed: 21-December-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_diracdelta, organization={Wolfram Research}, title={DiracDelta}, year={1999}, url={https://reference.wolfram.com/language/ref/DiracDelta.html}, note=[Accessed: 21-December-2024 ]}