represents the Dirac delta function .
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 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.
Examplesopen allclose all
Basic Examples (3)
DiracDelta vanishes for nonzero arguments:
DiracDelta stays unevaluated for :
Plot over a subset of the reals:
Use DiracDelta in an integral:
Numerical Evaluation (4)
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:
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:
DiracDelta is differentiable, but its derivative does not have a special name:
Differentiate the multivariate DiracDelta:
Differentiate a composition involving DiracDelta:
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:
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:
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:
Possible Issues (8)
Only HeavisideTheta gives DiracDelta after differentiation:
This also holds for the multivariate case:
DiracDelta 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:
Wolfram Research (1999), DiracDelta, Wolfram Language function, https://reference.wolfram.com/language/ref/DiracDelta.html.
Wolfram Language. 1999. "DiracDelta." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/DiracDelta.html.
Wolfram Language. (1999). DiracDelta. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/DiracDelta.html