Built-in Mathematica Symbol

DiracDelta

DiracDelta[x]
represents the Dirac delta function

.

DiracDelta[x₁, x₂, ...]
represents the multidimensional Dirac delta function

.

MORE INFORMATION

DiracDelta[x] returns 0 for all 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 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 stays unevaluated for

:

Canonicalize arguments:

DiracDelta stays unevaluated for

:

In[1]:=

Out[1]=

Canonicalize arguments:

In[1]:=

Out[1]=

Scope (4)

Integrate integrands containing DiracDelta over infinite and finite domains:

Integrate expressions containing derivatives of DiracDelta:

Interpret as the indefinite integral for real arguments:

TraditionalForm formatting:

Generalizations & Extensions (2)

Multivariate DiracDelta:

Differentiate the multivariate DiracDelta:

Applications (8)

Find classical harmonic oscillator Green function:

Solve the inhomogeneous ODE through convolution with the 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: