The packages making up the context
[Oppenheim and Willsky 1983].
The region of convergence for the Laplace transform is a strip, of the
form
The continuous-time Fourier transform is a function of a real variable
ω:
The corresponding commands are
<#566#>LaPlace<#566#> and <#567#>InvLaPlace<#567#> are more comprehensive than the
Laplace-transform commands provided with the 1.2 release of <#1166#>Mathematica<#1166#>,
which support only one-dimensional, one-sided Laplace transforms, and
do not keep track of the region of convergence.
Both versions of the Laplace transform, however, can be loaded into
<#1167#>Mathematica<#1167#> since the bilateral and unilateral transforms have different
naming conventions.
The continuous-time analogs of <#568#>DTFTransform<#568#>, <#569#>InvDTFTransform<#569#>,
and <#570#>ZSolve<#570#> are
The function
Finally, here is an example with Dirac delta functions.
The time-domain plot represents them with arrowheads,
a common engineering convention.
The value of the global variable $DeltaFunctionScaling indicates how
Delta functions are to be plotted: <#599#>None<#599#> means that all Delta functions
will have identical heights equal to the height of the plot, and
<#600#>Scaled<#600#> means that the height of each Delta function will be equal
to its area (the default).
This function has a Laplace transform but no poles or zeroes,
so a pole-zero diagram is not displayed.
<#601#>ASPAnalyze<#601#> does plot its magnitude and phase frequency responses,
although we have omitted them here.
<#571#>CTFTransform<#571#>, <#572#>InvCTFTransform<#572#>, and
<#573#>LSolve<#573#>.
The forward Fourier transform command <#574#>CTFTransform<#574#>
relies on <#575#>LaPlace<#575#>, and the inverse Fourier transform command
<#576#>InvCTFTransform<#576#> is built on top of <#577#>InvLaPlace<#577#>.
Because their definitions are very similiar, <#578#>CTFTransform<#578#> and
<#579#>InvCTFTransform<#579#> share the same rule base.
<#580#>LSolve<#580#> solves linear differential equations with constant
coefficients; it takes the equation and the unknown function as arguments
and the initial conditions and justification level as options.
For example, the differential equation
<#604#>PoleZeroPlot::notrational:<#604#>
<#606#>Transform is not a rational polynomial.<#606#>
<#607#>PoleZeroPlot::noplot:<#607#>
<#609#>A pole-zero plot cannot be generated.<#609#>
<#612#>ASPAnalyze::notinteresting:<#612#>
<#614#>Could not determine the important section of the frequency response.<#614#>