BUILT-IN MATHEMATICA SYMBOL

DiracDelta

DiracDelta[x]
represents the Dirac delta function

.

DiracDelta

represents the multidimensional Dirac delta function

.

MORE INFORMATION

DiracDelta[x] returns for all numeric x other than .

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 returns if any of the are numeric and not .

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

CLOSE ALL

Basic Examples (3)

DiracDelta stays unevaluated for

:

Canonicalize arguments:

In[1]:=

Out[1]=

In[2]:=

Out[2]=

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 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

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: