This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

Convolve

 Convolve gives the convolution with respect to x of the expressions f and g. Convolvegives the multidimensional convolution.
• The convolution of two functions and is given by .
• The multidimensional convolution is given by .
• The following options can be given:
 Assumptions \$Assumptions assumptions to make about parameters GenerateConditions False whether to generate conditions on parameters Method Automatic method to use PrincipalValue False whether to use principal value integrals
Convolve two functions:
A typical impulse response h for a system:
The step response corresponding to the same system:
Convolve two functions:
 Out[1]=
 Out[2]=

A typical impulse response h for a system:
 Out[2]=
The step response corresponding to the same system:
 Out[3]=
 Out[4]=
 Scope   (5)
The convolution gives the product integral of translates:
Elementary functions:
A convolution typically smooths the function:
For this family they all have unit area:
The convolution gives the product integral of translates:
Convolution with multivariate delta functions acts as a point operator:
Convolution with a function of bounded support acts as a filter:
Multiplication by UnitStep effectively gives the convolution on a finite interval:
 Options   (2)
Specify assumptions on a variable or parameter:
Generate conditions for the range of a parameter:
 Applications   (5)
Obtain a particular solution for a linear ordinary differential equation using convolution:
The convolution of UnitBox with itself is UnitTriangle:
Convolving the PDF of UniformDistribution with itself gives a TriangularDistribution:
UniformSumDistribution[n] is the convolution of n UniformDistribution PDFs:
ErlangDistribution is the convolution of k ExponentialDistribution[] PDFs:
Convolve computes an integral over the real line:
Convolution with DiracDelta gives the function itself:
Scaling:
Commutativity:
Distributivity:
The Laplace transform of a causal convolution is a product of the individual transforms:
The Fourier transform of a convolution is related to the product of the individual transforms:
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