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# DiracDelta

 DiracDelta[x]represents the Dirac delta function . DiracDelta[x1, x2, ...]represents the multidimensional Dirac delta function .
• 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 xi are numeric and not 0.
• 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.
DiracDelta stays unevaluated for :
Canonicalize arguments:
 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:
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:
Expand DiracDelta into DiracDelta with linear arguments:
Simplify expressions containing DiracDelta:
Fourier transforms:
Laplace transforms:
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:
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:
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