The piecewise convolution Notebook shows the student how to solve
one-dimensional continuous-time convolution problems using
the piecewise convolution package.
In the introduction, the Notebook informs the user that a piecewise
function can be represented as a list of intervals.
Each interval is itself a list containing the start of the interval,
the end of the interval, and the function defined on that interval.
For example, the function
or in list form as
The Notebook gives several examples of piecewise convolution using
the newly defined object <#671#>PiecewiseConvolution<#671#>, which
requires three arguments--- f (t), g(t), and t.
For example, the <#1184#>Mathematica<#1184#> dialogue
illustrates the shifting property of the Dirac delta function.
We could have used the piecewise form of
<#674#>Delta[t - a]<#674#>, which is <#675#>{Area[1], a, a}<#675#>, in this example.
The Notebook also gives an example of a convolution that produces an
unstable function:
Part of computing a piecewise convolution requires identifying the
overlapping intervals by drawing pictures of f (u) and g(t - u).
So, for two simple functions,
the Notebook shows an <#677#>animation<#677#> of g(t - u) sliding through
f (u).
Each frame shows an important interval over which to integrate, and
the reader can control the frame rate.
<#676#>Integrate::divg: Integral does not converge<#676#>