(*^

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:[font = title; inactive; ]
Testing the Laplace Transform
Capabilities of the 
Signal Processing Packages
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by Brian L. Evans and James H. McClellan
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Digital Signal Processing Laboratory
School of Electrical Engineering
Georgia Institute of Technology
Atlanta, GA  30332-0250

EMAIL:  evans@eedsp.gatech.edu
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		This notebook tests the forward bilateral Laplace transform LaPlace and the inverse bilateral Laplace transform InvLaPlace.  These Laplace transform rule bases are a part of the analog signal processing packages.  A user, however, can use the Laplace transform rule bases without possessing any knowledge of signal processing.
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		This notebook should have automatically loaded the analog signal processing packages (see the Initialization section).   If Laplace transforms will be used frequently, then it might be a good idea to rebuild the kernel so that it includes the analog signal packages (see Mathematica's Dump command), since this loading process does take a few minutes under Mathematica 1.2.
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		One advantage of these Laplace rule bases is that the user can ask it (via the Dialogue option) to document the process by which it computes the transform.  Namely, LaPlace and InvLaPlace can completely justify their answers.
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		Remember that to expand a group of cells, just move the mouse to the downward pointing half arrow and double click.  In general, we have grouped cells so that each grouping will only occupy one screen (or less) if the notebook window is resized so that it spans the heighth of the screen.
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This notebook is not a tutorial on the
Laplace transform.
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Initialization
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*)
Needs[ "SignalProcessing`Analog`" ]
(*
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Loading version 2.32 of the signal processing support\
 
  packages ...
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Support module has been loaded.
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The knowledge base of signal processing operators has\
 
  been loaded.
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Supporting routines for filter design are loaded.
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Region of convergence routines for the transform rule\
 
  bases are loaded.
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Piecewise convolution rules have been loaded.
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Supporting routines for transform rule bases are loaded.
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Support::loaded: 
   Supporting routines, objects, and rules for the signal
    processing package have been loaded successfully.
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Support::change: 
   The loading process has changed such that the files
    Tree.m and DataType.m are no longer automatically
    loaded (the signal processing packages no longer rely
    on the definitions in these files).
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Loading version 2.32 of the analog signal processing\
 
  packages ...
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The one-dimensional analog filter design objects are\
 
  loaded.
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Supporting routines and objects for the Laplace transform\
 
  are loaded.
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The forward Laplace transform rule base LaPlace has been\
 
  loaded.
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The inverse Laplace transform rule base InvLaPlace has\
 
  been loaded.
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The supporting functions are loaded for symbolic Fourier\
 
  transforms.
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The continuous Fourier transform (CTFT) functions are\
 
  loaded.
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LSolve, a differential equation solver, is loaded.
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The analog signal analyzer is now loaded.
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Analog::loaded: 
   Routines, objects, and rules for the analog signal
    processing packages have been loaded successfully.
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Analog signal processing extensions have been loaded.
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Good starting points are the objects ASPAnalyze and\
 
  LaPlace.
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Active packages are maintained in the variable\
 
  $ContextPath.
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  New signal primitives are in SPsignals.
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  New system primitives are in SPoperators.
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  New Mathematica functions are in SPfunctions.
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Note that all signal processing expressions will
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be automatically simplified.
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References
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Ruel Churchill.  Operational Mathematics.  McGraw-Hill, NY.  1958.
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Fritz Oberhettinger and Larry Badii.  Tables of Laplace Transforms.  1973.
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Alan Oppenheim and Allan Willsky.  Signals and Systems.  Prentice Hall, Englewood Cliffs (NJ).  1983.
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Forward Laplace Transforms
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		The forward Laplace transform rule base transforms a two-sided continuous-time function to the Laplace domain.  In these packages, the Laplace transform of the continuous-time function f(t) is F(s) = L{f(t)} such that
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In order that a transform's inverse can be specified uniquely, LaPlace tracks the region of convergence of the transform.
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Invoking the Forward Laplace Transform Rule Base
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		To invoke LaPlace, a user must specify the function and the time variable(s) to the rule base.  For example, the two-sided Laplace transform of f(t) = 1 is Delta[s]:
:[font = input; startGroup; ]

LaPlace[ 1, t ]
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LTransData[Delta[s], Rminus[0], Rplus[0], LVariables[s]]
;[o]
LTransData[Delta[s], Rminus[0], Rplus[0], LVariables[s]]
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Notice that the Laplace transform rule base returns an object with a head of LTransData.  The slots of LTransData are the Laplace transform function, its region of convergence (two slots), and the Laplace transform variables.  In this case, LaPlace used "s" as the transform variable (the default). 
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Two-sided and one-sided transforms
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		The Laplace transform assumes that the time function is two-sided.  In order to specify a one-sided time function, the continuous step function CStep should appear in each multiplicative term.  For example, the Laplace transform of f(t) = { 1 for t > 0, 1/2 for t = 0, and 0 elsewhere } is 1/s:
:[font = input; startGroup; ]

LaPlace[ CStep[t], t, s ]
:[font = output; inactive; output; endGroup; ]
LTransData[s^(-1), Rminus[0], 
  Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
           1
LTransData[-, Rminus[0], Rplus[Infinity], 
           s
 
  LVariables[s]]
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CStep[t] is the continuous step function.  The left-sided version of  this function, CStep[-t], has the following Laplace transform:
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LaPlace[ CStep[-t], t, s ]
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LTransData[-s^(-1), Rminus[DirectedInfinity[-1]], 
  Rplus[0], LVariables[s]]
;[o]
             1
LTransData[-(-), Rminus[-Infinity], Rplus[0], 
             s
 
  LVariables[s]]
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Optional arguments (options)
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		If a user supplies the Laplace transform variable, then the user can also specify options.  The default options are:
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Options[LaPlace]
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{Dialogue -> True, Simplify -> True, 
  TransformLookup -> {}}
;[o]
{Dialogue -> True, Simplify -> True, 
 
  TransformLookup -> {}}
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-- Dialogue Option
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		If set to True or All, the Dialogue setting will cause LaPlace to report any assumptions being made about the parameters in the continuous-time function.  It will also report those functions which it could not transform.  Proper usage of the Dialogue option is:
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LaPlace[ Exp[t], t, s, Dialogue -> True ]
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Transform::incomplete: 
   The rule base could not compute the 
                                  t
    forward Laplace transform of E  with respect to t.
:[font = output; inactive; output; endGroup; endGroup; ]
MakeLObject[SignalProcessing`Analog`LaPlace`Private`mylapla\
    ce[E^t, t, s, {True, True, False, False, False, False, 
    False}, {t}, {s}, {Dialogue -> True, Dialogue -> True, 
    Simplify -> True}], s]
;[o]
-Incomplete Laplace Transform-
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In this case, we have asked for the two-sided Laplace transform of Exp[t], which does not exist.  Since Dialogue is enabled (which is the default anyway), LaPlace reports that it cannot find the transform of Exp[t].  Whether Dialogue is enabled or not, the Laplace rule base will return an invalid Laplace transform object for those forward transform which do not exist.
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-- TransformLookup Option
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		The TransformLookup option allows users to specify their own transform pairs.  One consequence of this is that users can now to transforms of general functions like x[t] without having to define x[t] as a formula.  For example,
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LaPlace[ a x[t], t, s, TransformLookup -> {x[t] :> X[s]} ]
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LTransData[a*X[s], Rminus[0], 
  Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
LTransData[a X[s], Rminus[0], Rplus[Infinity], 
 
  LVariables[s]]
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Here, we have let the Laplace transform of x[t] be X[s].
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-- Simplify Option
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		The other option is Simplify.  If true, the signal processing simplification rules (SPSimplificationRules) are applied to the entire transform object (including the region of convergence information).   This rule base reduces expressions involving the abs, conj, imag, max, min, and real operators in Mathematica (Abs, Conjugate, Im, Max, Min, and Re, respectively).  The LaPlace rule base runs faster when this option is disabled, but the expressions for the limits on the region of convergence will not be simplified.
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Organization of the Forward Laplace Transform Rule Base
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		The forward Laplace transform rule base transforms one-dimensional and multidimensional expressions.  The dimension of the transform is equal to the number of  time variables passed to LaPlace.  For each dimension, LaPlace calls a one-dimensional rule base is called.  The one-dimensional rule base, MyLaPlace, is an ordered collection of 95 rules:
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*)
Length[SignalProcessing`Analog`LaPlace`Private`LaPlaceRules]
(*
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97
;[o]
97
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These rules are grouped into six sections:
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MatrixForm[{"I.   multidimensional hooks     3 rules", 
   "II.  rational transform pairs  13 rules", 
   "III. non-rat. transform pairs  38 rules", 
   "IV.  transform properties      20 rules", 
   "V.   transforms of operators   12 rules", 
   "VI.  transform strategies       9 rules"}]
;[o]
I.   multidimensional hooks     3 rules
 
II.  rational transform pairs  13 rules
 
III. non-rat. transform pairs  38 rules
 
IV.  transform properties      20 rules
 
V.   transforms of operators   12 rules
 
VI.  transform strategies       9 rules
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The multidimensional hooks are only necessary to maintain the region of convergence for non-separable multidimensional functions.  For the most part, the transform pairs and properties are adapted from tables in Bracewell's {The Fourier Transform and Its Applications} and Churchill's {Operational Mathematics}.   The Laplace transforms of operators were adapted from these sources as well as from {Tables of Laplace Transforms} by Oberhettinger and Badii.  The strategies were ad hoc rules to massage expressions into the right form for transformation.
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Rational One-Dimensional Transforms
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		The Laplace transform of f(t) = 1 for t > 0  is rational.  That is, the Laplace transform function is a rational polynomial.
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		The Laplace rule base transforms time functions by using lookup tables.  The following four transforms require the use of one Laplace transform property --  they all use the fact that multiplying f(t) by an exponential function is the transform of f(t) shifted by -a.  The last two also invoke the additivity property.
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Transforms requiring only one property
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LaPlace[ t Exp[a t] CStep[t], t, s ]
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LTransData[(-a + s)^(-2), Rminus[Real[a]], Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
                   -2
LTransData[(-a + s)  , Rminus[Real[a]], Rplus[Infinity], LVariables[s]]
:[font = input; startGroup; ]

LaPlace[ t^(n-1) Exp[a t] CStep[t] / Gamma[n], t, s ]
:[font = output; inactive; output; endGroup; ]
LTransData[(-a + s)^(-n), Rminus[Real[a]], Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
                   -n
LTransData[(-a + s)  , Rminus[Real[a]], Rplus[Infinity], LVariables[s]]
:[font = input; startGroup; ]

LaPlace[ ( Exp[a t] - Exp[b t] ) CStep[t] / ( a - b ), t, s ]
:[font = output; inactive; output; endGroup; ]
LTransData[1/((-a + s)*(-b + s)), Rminus[Max[Real[a], Real[b]]], Rplus[DirectedInfinity[1]], 
  LVariables[s]]
;[o]
                   1
LTransData[-----------------, Rminus[Max[Real[a], Real[b]]], Rplus[Infinity], LVariables[s]]
           (-a + s) (-b + s)
:[font = input; startGroup; ]

LaPlace[ ( a Exp[a t] - b Exp[b t] ) CStep[t] / ( a - b ), t, s ]
:[font = output; inactive; output; endGroup; endGroup; ]
LTransData[s/((-a + s)*(-b + s)), Rminus[Max[Real[a], Real[b]]], Rplus[DirectedInfinity[1]], 
  LVariables[s]]
;[o]
                   s
LTransData[-----------------, Rminus[Max[Real[a], Real[b]]], Rplus[Infinity], LVariables[s]]
           (-a + s) (-b + s)
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Four common rational transforms
:[font = input; startGroup; ]
LaPlace[ t CStep[t], t, s ]
:[font = output; inactive; output; endGroup; ]
LTransData[s^(-2), Rminus[0], Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
            -2
LTransData[s  , Rminus[0], Rplus[Infinity], LVariables[s]]
:[font = input; startGroup; ]
LaPlace[ t CStep[t] - CStep[t], t, s ]
:[font = output; inactive; output; endGroup; ]
LTransData[-((-1 + s)/s^2), Rminus[0], 
  Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
             -1 + s
LTransData[-(------), Rminus[0], Rplus[Infinity], 
                2
               s
 
  LVariables[s]]
:[font = input; startGroup; ]

LaPlace[ t^(n - 1) CStep[t] / Gamma[n], t, s ]
:[font = output; inactive; output; endGroup; ]
LTransData[s^(-n), Rminus[0], Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
            -n
LTransData[s  , Rminus[0], Rplus[Infinity], LVariables[s]]
:[font = input; startGroup; ]

LaPlace[ Exp[a t] CStep[t], t, s ]
:[font = output; inactive; output; endGroup; endGroup; ]
LTransData[(-a + s)^(-1), Rminus[Real[a]], Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
             1
LTransData[------, Rminus[Real[a]], Rplus[Infinity], LVariables[s]]
           -a + s
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Hyperbolic Functions
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		Mathematica automatically translates Sinh[a t] into ( - Exp[- a t] + Exp[a t] ) / 2 so that the Laplace transform rule base does not need a lookup rule for the hyperbolic sine.  This is also true for Cosh[a t], which is always rewritten as ( Exp[- a t] + Exp[a t] ) / 2.
:[font = input; startGroup; ]

LaPlace[ Sinh[a t] CStep[t] / a, t, s ]
:[font = output; inactive; output; endGroup; ]
LTransData[(-a^2 + s^2)^(-1), Rminus[Max[-Re[a], Re[a]]], 
  Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
              1
LTransData[--------, Rminus[Max[-Re[a], Re[a]]], 
             2    2
           -a  + s
 
  Rplus[Infinity], LVariables[s]]
:[font = input; startGroup; ]

LaPlace[ Cosh[a t] CStep[t], t, s ]
:[font = output; inactive; output; endGroup; endGroup; ]
LTransData[s/(-a^2 + s^2), Rminus[Max[-Re[a], Re[a]]], 
  Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
              s
LTransData[--------, Rminus[Max[-Re[a], Re[a]]], 
             2    2
           -a  + s
 
  Rplus[Infinity], LVariables[s]]
:[font = subsubsection; inactive; startGroup; Cclosed; ]
Transforms of Sinusoids
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		Here are some examples of  transforms of  terms involving only sinusoidal functions:

:[font = input; startGroup; ]
LaPlace[ Sin[a t] CStep[t] / a, t, s ]
:[font = output; inactive; output; endGroup; ]
LTransData[(a^2 + s^2)^(-1), Rminus[0], 
  Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
              1
LTransData[-------, Rminus[0], Rplus[Infinity], 
            2    2
           a  + s
 
  LVariables[s]]
:[font = input; startGroup; ]

LaPlace[ Cos[a t] CStep[t], t, s ]
:[font = output; inactive; output; endGroup; ]
LTransData[s/(a^2 + s^2), Rminus[0], 
  Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
              s
LTransData[-------, Rminus[0], Rplus[Infinity], 
            2    2
           a  + s
 
  LVariables[s]]
:[font = input; startGroup; ]

LaPlace[ ( 1 - Cos[a t] ) CStep[t] / a^2, t, s ]
:[font = output; inactive; output; endGroup; ]
LTransData[(a^2*s + s^3)^(-1), Rminus[0], 
  Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
               1
LTransData[---------, Rminus[0], Rplus[Infinity], 
            2      3
           a  s + s
 
  LVariables[s]]
:[font = input; startGroup; ]

LaPlace[ ( a t - Sin[a t] ) CStep[t] / a^3, t, s ]
:[font = output; inactive; output; endGroup; ]
LTransData[(a^2*s^2 + s^4)^(-1), Rminus[0], 
  Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
               1
LTransData[----------, Rminus[0], Rplus[Infinity], 
            2  2    4
           a  s  + s
 
  LVariables[s]]
:[font = input; startGroup; ]

LaPlace[ ( Sin[a t] - a t Cos[a t] ) CStep[t] / ( 2 a^3 ), t, s ]
:[font = output; inactive; output; endGroup; ]
LTransData[(a^2 + s^2)^(-2), Rminus[0], 
  Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
             2    2 -2
LTransData[(a  + s )  , Rminus[0], Rplus[Infinity], 
 
  LVariables[s]]
:[font = input; startGroup; ]

LaPlace[ t Sin[a t] CStep[t] / ( 2 a ), t, s ]
:[font = output; inactive; output; endGroup; ]
LTransData[s/(a^2 + s^2)^2, Rminus[0], 
  Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
               s
LTransData[----------, Rminus[0], Rplus[Infinity], 
             2    2 2
           (a  + s )
 
  LVariables[s]]
:[font = input; startGroup; ]

LaPlace[ t Cos[a t] CStep[t], t, s ]
:[font = output; inactive; output; endGroup; endGroup; ]
LTransData[-(((a - s)*(a + s))/(a^2 + s^2)^2), Rminus[0], 
  Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
             (a - s) (a + s)
LTransData[-(---------------), Rminus[0], Rplus[Infinity], 
                 2    2 2
               (a  + s )
 
  LVariables[s]]
:[font = subsubsection; inactive; startGroup; Cclosed; ]
Sinusoids Mixed With Other Functions
:[font = text; inactive; ]

		The next three examples involving continuous-time functions which are a mixture of sinusoidal and exponential terms.  The first two examples are damped sinusoids.
:[font = input; startGroup; ]

LaPlace[ Exp[a t] Sin[b t] CStep[t] / b, t, s ]
:[font = output; inactive; output; endGroup; ]
LTransData[(b^2 + (-a + s)^2)^(-1), Rminus[Re[a]], 
  Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
                 1
LTransData[--------------, Rminus[Re[a]], 
            2           2
           b  + (-a + s)
 
  Rplus[Infinity], LVariables[s]]
:[font = input; startGroup; ]

LaPlace[ Exp[a t] Cos[b t] CStep[t], t, s ]
:[font = output; inactive; output; endGroup; ]
LTransData[(-a + s)/(b^2 + (-a + s)^2), Rminus[Re[a]], 
  Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
               -a + s
LTransData[--------------, Rminus[Re[a]], Rplus[Infinity], 
            2           2
           b  + (-a + s)
 
  LVariables[s]]
:[font = input; startGroup; ]

LaPlace[ ( Sin[a t] Cosh[a t] - Cos[a t] Sinh[a t] ) CStep[t],
		 t, s ]
:[font = output; inactive; output; endGroup; endGroup; ]
LTransData[(4*a^3)/(4*a^4 + s^4), 
  Rminus[Max[-Re[a], Re[a]]], Rplus[DirectedInfinity[1]], 
  LVariables[s]]
;[o]
                3
             4 a
LTransData[---------, Rminus[Max[-Re[a], Re[a]]], 
              4    4
           4 a  + s
 
  Rplus[Infinity], LVariables[s]]
:[font = subsubsection; inactive; startGroup; Cclosed; ]
Other Rational Transforms
:[font = text; inactive; ]

		For the most part, rational transforms represent some combination of continuous-time exponential and sinusoidal functions.  Another example of this is the forward Laplace transform of a Laguerre polynomial of order n:
:[font = input; startGroup; ]
LaPlace[ LaguerreL[n, t] CStep[t], t, s ]
:[font = output; inactive; output; endGroup; ]
LTransData[(-1 + s)^n*s^(-1 - n), Rminus[0], 
  Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
                   n  -1 - n
LTransData[(-1 + s)  s      , Rminus[0], Rplus[Infinity], 
 
  LVariables[s]]
:[font = text; inactive; ]

One rational transform pair which does not contain sinusoidal or exponential terms is:
:[font = input; startGroup; ]

LaPlace[ (t / (2 a))^(k - 1/2) BesselJ[k - 1/2, a t] CStep[t] /
		  Gamma[k], t, s ]
:[font = output; inactive; output; endGroup; endGroup; endGroup; ]
LTransData[1/(Pi^(1/2)*(a^2 + s^2)^k), Rminus[0], 
  Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
                    1
LTransData[-------------------, Rminus[0], Rplus[Infinity], 
                      2    2 k
           Sqrt[Pi] (a  + s )
 
  LVariables[s]]
:[font = subsection; inactive; startGroup; Cclosed; ]
Non-rational One-Dimensional Transforms
:[font = text; inactive; ]

		More transform pairs exist for non-rational transforms than for rational ones.  In general, these non-rational transform pairs involve continuous-time Bessel and error functions as well as those of the continuous-time cosine, sine, and exponential integrals:
		
								Conventional							Mathematica
									Notation																	Code

									Ci	(t)																	CosIntegral	[t]	
									Si	(t)																	SinIntegral[t]
									Ei (t)																ExpIntegralEi[t]
									Erf(t)															Erf[t]	

									In(t)																	BesselI[n, t]																		
									Jn(t)																	BesselJ[n, t]
									Kn(t)															BesselK[n, t]
									Ln(t)																BesselL[n, t]


Here are several transform pairs involving these functions:
:[font = input; startGroup; ]

LaPlace[ ( 1 / Sqrt[Pi t] + a Exp[a^2 t] Erf[a Sqrt[t]] ) CStep[t],
		 t, s ]
:[font = output; inactive; output; endGroup; ]
LTransData[s^(1/2)/(-a^2 + s), Rminus[Max[0, Re[a^2]]], 
  Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
           Sqrt[s]                    2
LTransData[-------, Rminus[Max[0, Re[a ]]], Rplus[Infinity], 
             2
           -a  + s
 
  LVariables[s]]
:[font = input; startGroup; ]

LaPlace[ -2 CosIntegral[t / k] CStep[t], t, s ]
:[font = output; inactive; output; endGroup; ]
LTransData[Log[1 + k^2*s^2]/s, Rminus[0], 
  Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
                    2  2
           Log[1 + k  s ]
LTransData[--------------, Rminus[0], Rplus[Infinity], 
                 s
 
  LVariables[s]]
:[font = input; startGroup; ]

LaPlace[ ( 2 Log[Pi] - 2 CosIntegral[Pi t] ) CStep[t], t, s ]
:[font = output; inactive; output; endGroup; ]
LTransData[Log[Pi^2 + s^2]/s, Rminus[0], 
  Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
                 2    2
           Log[Pi  + s ]
LTransData[-------------, Rminus[0], Rplus[Infinity], 
                 s
 
  LVariables[s]]
:[font = input; startGroup; ]

LaPlace[ CosIntegral[a t + b] CStep[t], t, s ]
:[font = output; inactive; output; endGroup; ]
LTransData[-(E^((b*s)/a)*a*Log[1 + s^2/a^2])/(2*s*Abs[a]), Rminus[0], 
  Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
                                 2
              (b s)/a           s
           -(E        a Log[1 + --])
                                 2
                                a
LTransData[-------------------------, Rminus[0], Rplus[Infinity], 
                  2 s Abs[a]
 
  LVariables[s]]
:[font = input; startGroup; ]

LaPlace[ - ExpIntegralEi[- t / k] CStep[t], t, s ]
:[font = output; inactive; output; endGroup; ]
LTransData[Log[1 + k*s]/s, Rminus[0], 
  Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
           Log[1 + k s]
LTransData[------------, Rminus[0], Rplus[Infinity], 
                s
 
  LVariables[s]]
:[font = input; startGroup; ]

LaPlace[ Sin[k t] CStep[t] / t, t, s ]
:[font = output; inactive; output; endGroup; ]
LTransData[ArcTan[k/s], Rminus[Abs[Im[k]]], 
  Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
                  k
LTransData[ArcTan[-], Rminus[Abs[Im[k]]], Rplus[Infinity], 
                  s
 
  LVariables[s]]
:[font = text; inactive; ]
Alternatively,
:[font = input; startGroup; ]
LaPlace[ k Sinc[k t] CStep[t], t, s ]
:[font = output; inactive; output; endGroup; ]
LTransData[(k*ArcTan[k/s])/Abs[k], Rminus[0], 
  Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
                    k
           k ArcTan[-]
                    s
LTransData[-----------, Rminus[0], Rplus[Infinity], 
             Abs[k]
 
  LVariables[s]]
:[font = input; startGroup; ]

LaPlace[ SinIntegral[k t + b] CStep[t], t, s ]
:[font = output; inactive; output; endGroup; ]
LTransData[(E^((b*s)/k)*ArcTan[k/s])/s, Rminus[0], 
  Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
            (b s)/k        k
           E        ArcTan[-]
                           s
LTransData[------------------, Rminus[0], Rplus[Infinity], 
                   s
 
  LVariables[s]]
:[font = input; startGroup; ]

LaPlace[ CStep[t - k] / Sqrt[Pi t], t, s ]   (* incorrect *)
:[font = output; inactive; output; endGroup; ]
LTransData[s^(-1/2), Rminus[0], Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
              1
LTransData[-------, Rminus[0], Rplus[Infinity], LVariables[s]]
           Sqrt[s]
:[font = input; startGroup; ]

LaPlace[ Sin[2 k Sqrt[t]] CStep[t] / ( Pi t ), t, s ]
:[font = output; inactive; output; endGroup; ]
LTransData[Erf[k/s^(1/2)], Rminus[DirectedInfinity[-1]], 
  Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
                  k
LTransData[Erf[-------], Rminus[-Infinity], Rplus[Infinity], 
               Sqrt[s]
 
  LVariables[s]]
:[font = input; startGroup; ]

LaPlace[ BesselI[n, a t] CStep[t], t, s ]
:[font = output; inactive; output; endGroup; ]
LTransData[(s - (-a^2 + s^2)^(1/2))^n/
   (a^n*(-a^2 + s^2)^(1/2)), Rminus[Abs[Re[a]]], 
  Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
                       2    2  n
           (s - Sqrt[-a  + s ])
LTransData[---------------------, Rminus[Abs[Re[a]]], 
              n        2    2
             a  Sqrt[-a  + s ]
 
  Rplus[Infinity], LVariables[s]]
:[font = input; startGroup; ]

LaPlace[ 2 BesselK[0, 2 Sqrt[2 k t]] CStep[t] / Sqrt[Pi t], t, s ]
:[font = output; inactive; output; endGroup; ]
LTransData[(E^(k/s)*BesselK[0, k/s])/s^(1/2), Rminus[0], 
  Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
            k/s            k
           E    BesselK[0, -]
                           s
LTransData[------------------, Rminus[0], Rplus[Infinity], 
                Sqrt[s]
 
  LVariables[s]]
:[font = input; startGroup; ]

LaPlace[ HermiteH[3, Sqrt[t]] CStep[t] / ( 3! Sqrt[Pi] ), t, s ]
:[font = output; inactive; output; endGroup; endGroup; ]
LTransData[(1 - s)/s^(5/2), Rminus[0], Rplus[DirectedInfinity[1]], 
  LVariables[s]]
;[o]
           1 - s
LTransData[-----, Rminus[0], Rplus[Infinity], LVariables[s]]
            5/2
           s
:[font = subsection; inactive; startGroup; Cclosed; ]
Properties
:[font = text; inactive; ]

		The LaPlace rule base implements the following properties:

Additivity																							L{a(t) + b(t)}			 = L{a(t)} + L{b(t)}
Homogeneity																						L{c a(t)}												= c L{a(t)
Shift-in-Time																						L{f(t - c)}											= exp(- c s) L{f(t)}
Multiplication-by-Exponential						exp(c t) L{f(t)}		= L{f(t)}, s -> s - c
Multiplication-by-Time																		L{t f(t)}														= - d/ds L{f(t)}
Sinusoidal Modulation																							
Reversal-in-Time																				L{f(-t)}															= L{f(t)}, s -> -s
Similarity																							L{f(c t}}													= L{f(t)} / |c|, s -> s/c
:[font = subsubsection; inactive; startGroup; Cclosed; ]
Additivity
:[font = text; inactive; ]

		The Laplace transform of a(t) + b(t) is A(s), the Laplace transform of a(t), plus B(s), the Laplace transform of b(t).  The new region of convergence contains the intersection of the region of convergence of A(s) and B(s):
:[font = input; startGroup; ]

LaPlace[ Delta[t] + CStep[t - 1], t, s, Dialogue -> All ]
:[font = print; inactive; startGroup; ]
L  {CStep[-1 + t] + Delta[t]}
  t
which becomes
L  {CStep[-1 + t]} + L  {Delta[t]}
  t                    t
which becomes
L  {CStep[t]}
  t
------------- + Transform[1, -Infinity, Infinity]
      s
     E
which becomes
               1
Transform[1 + ----, 0, Infinity]
               s
              E  s
which becomes
               1
Transform[1 + ----, 0, Infinity]
               s
              E  s
:[font = output; inactive; output; endGroup; endGroup; endGroup; ]
LTransData[1 + 1/(E^s*s), Rminus[0], 
  Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
                1
LTransData[1 + ----, Rminus[0], Rplus[Infinity], 
                s
               E  s
 
  LVariables[s]]
:[font = subsubsection; inactive; startGroup; Cclosed; ]
Homogeneity
:[font = text; inactive; ]

		This property moves all terms not dependent on the time variable outside of the Laplace transform operator with no effect on the region of convergence:

:[font = input; startGroup; ]
LaPlace[ A K CStep[t] / 2, t, s, Dialogue -> All ]
:[font = print; inactive; startGroup; ]
    A K CStep[t]
L  {------------}
  t      2
which becomes
A K L  {CStep[t]}
      t
-----------------
        2
which becomes
          A K
Transform[---, 0, Infinity]
          2 s
which becomes
          A K
Transform[---, 0, Infinity]
          2 s
:[font = output; inactive; output; endGroup; endGroup; endGroup; ]
LTransData[(A*K)/(2*s), Rminus[0], 
  Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
           A K
LTransData[---, Rminus[0], Rplus[Infinity], LVariables[s]]
           2 s
:[font = subsubsection; inactive; startGroup; Cclosed; ]
Shift-in-Time
:[font = input; startGroup; ]

LaPlace[ Exp[t] CStep[t], t, s ]
:[font = output; inactive; output; endGroup; ]
LTransData[(-1 + s)^(-1), Rminus[1], 
  Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
             1
LTransData[------, Rminus[1], Rplus[Infinity], 
           -1 + s
 
  LVariables[s]]
:[font = input; startGroup; ]
LaPlace[ Exp[t - 7] CStep[t - 7], t, s ]
:[font = output; inactive; output; endGroup; endGroup; ]
LTransData[1/(E^(7*s)*(-1 + s)), Rminus[1], 
  Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
                 1
LTransData[-------------, Rminus[1], Rplus[Infinity], 
            7 s
           E    (-1 + s)
 
  LVariables[s]]
:[font = subsubsection; inactive; startGroup; Cclosed; ]
Multiplication-by-Exponential
:[font = text; inactive; ]

		The multiplication of f(t) by exp(a t)  shifts F(s), the Laplace transform of  f(t), to the right by a, and the region of convergence is shifted by the real part of a:
:[font = input; startGroup; ]

LaPlace[ Cos[t] CStep[t], t, s ]
:[font = output; inactive; output; endGroup; ]
LTransData[s/(1 + s^2), Rminus[0], 
  Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
             s
LTransData[------, Rminus[0], Rplus[Infinity], 
                2
           1 + s
 
  LVariables[s]]
:[font = input; startGroup; ]

LaPlace[ Exp[a t] Cos[t] CStep[t], t, s ]
:[font = output; inactive; output; endGroup; endGroup; ]
LTransData[(-a + s)/(1 + (-a + s)^2), Rminus[Re[a]], 
  Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
              -a + s
LTransData[-------------, Rminus[Re[a]], Rplus[Infinity], 
                       2
           1 + (-a + s)
 
  LVariables[s]]
:[font = subsubsection; inactive; startGroup; Cclosed; ]
Multiplication-by-Time 
:[font = text; inactive; ]

		Multiplication in time corresponds to differentiation in the Laplace domain with no effect on the region of convergence:
:[font = input; startGroup; ]

LaPlace[ Exp[a t] CStep[t], t, s ]
:[font = output; inactive; output; endGroup; ]
LTransData[(-a + s)^(-1), Rminus[Re[a]], 
  Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
             1
LTransData[------, Rminus[Re[a]], Rplus[Infinity], 
           -a + s
 
  LVariables[s]]
:[font = input; startGroup; ]

LaPlace[ t Exp[a t] CStep[t], t, s ]
:[font = output; inactive; output; endGroup; endGroup; ]
LTransData[(-a + s)^(-2), Rminus[Re[a]], 
  Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
                   -2
LTransData[(-a + s)  , Rminus[Re[a]], Rplus[Infinity], 
 
  LVariables[s]]
:[font = subsubsection; inactive; startGroup; Cclosed; ]
Sinusoidal Modulation
:[font = input; startGroup; ]

LaPlace[ BesselJ[v, a t] CStep[t], t, s ]
:[font = output; inactive; output; endGroup; ]
LTransData[(-s + (a^2 + s^2)^(1/2))^v/
   (a^v*(a^2 + s^2)^(1/2)), Rminus[Abs[Im[a]]], 
  Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
                       2    2  v
           (-s + Sqrt[a  + s ])
LTransData[---------------------, Rminus[Abs[Im[a]]], 
              v       2    2
             a  Sqrt[a  + s ]
 
  Rplus[Infinity], LVariables[s]]
:[font = input; startGroup; ]
LaPlace[ Sin[a t] BesselJ[v, a t] CStep[t], t, s ]
:[font = output; inactive; output; endGroup; endGroup; ]
LTransData[I/2*(-((I*a - s + (-2*I*a*s + s^2)^(1/2))^v/
        (a^v*(-2*I*a*s + s^2)^(1/2))) + 
     (-I*a - s + (2*I*a*s + s^2)^(1/2))^v/
      (a^v*(2*I*a*s + s^2)^(1/2))), Rminus[Abs[Im[a]]], 
  Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
                                            2  v
           I    (I a - s + Sqrt[-2 I a s + s ])
LTransData[- (-(--------------------------------) + 
           2          v                  2
                     a  Sqrt[-2 I a s + s ]
 
                                 2  v
     (-I a - s + Sqrt[2 I a s + s ])
     --------------------------------), Rminus[Abs[Im[a]]], 
           v                 2
          a  Sqrt[2 I a s + s ]
 
  Rplus[Infinity], LVariables[s]]
:[font = subsubsection; inactive; startGroup; Cclosed; ]
Reversal-in-Time
:[font = input; startGroup; ]

LaPlace[ Exp[t] CStep[t], t, s ]
:[font = output; inactive; output; endGroup; ]
LTransData[(-1 + s)^(-1), Rminus[1], 
  Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
             1
LTransData[------, Rminus[1], Rplus[Infinity], 
           -1 + s
 
  LVariables[s]]
:[font = input; startGroup; ]

LaPlace[ Exp[-t] CStep[-t], t, s ]
:[font = output; inactive; output; endGroup; ]
LTransData[(-1 - s)^(-1), Rminus[DirectedInfinity[-1]], 
  Rplus[-1], LVariables[s]]
;[o]
             1
LTransData[------, Rminus[-Infinity], Rplus[-1], 
           -1 - s
 
  LVariables[s]]
:[font = input; startGroup; ]
LaPlace[ Exp[-t] CStep[t], t, s ]
:[font = output; inactive; output; endGroup; endGroup; ]
LTransData[(1 + s)^(-1), Rminus[-1], 
  Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
             1
LTransData[-----, Rminus[-1], Rplus[Infinity], 
           1 + s
 
  LVariables[s]]
:[font = subsubsection; inactive; startGroup; Cclosed; ]
Similarity
:[font = input; startGroup; ]

LaPlace[ Delta[t], t, s ]
:[font = output; inactive; output; endGroup; ]
LTransData[1, Rminus[DirectedInfinity[-1]], 
  Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
LTransData[1, Rminus[-Infinity], Rplus[Infinity], 
 
  LVariables[s]]
:[font = input; startGroup; ]

LaPlace[ Delta[a t], t, s ]
:[font = output; inactive; output; endGroup; endGroup; endGroup; ]
LTransData[Abs[a]^(-1), Rminus[DirectedInfinity[-1]], 
  Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
             1
LTransData[------, Rminus[-Infinity], Rplus[Infinity], 
           Abs[a]
 
  LVariables[s]]
:[font = subsection; inactive; startGroup; Cclosed; ]
Transforms of Operators
:[font = text; inactive; ]
		The LaPlace rule base also transforms these operators:

Absolute Value			L{ f(abs(t)) } = L{ f(t) u(t) } + L{ f(-t) u(-t) }
Conjugation				L{ Conjugate{f(t)} } = Conjugate{ F( Conjugate(s) ) }
Convolution			    L{ Convolve   {a(t),  b(t)} } = A(s) B(s)
																												t
Derivative					L{ d/dt a(t) } = s F(s) - f(+0)
										    t
Integral						L{ 	/   f(r) dr } =  F(s) / s 
											0
Imaginary					L{ Im{f(t)} } =
																																															P
Periodic						L{ Periodic							f(t) }  = /  exp(-s t) f(t) dt   /  ( 1- exp(-P s) )
																									P, t														0
Real								L{ Re{f(t)} } =
Reverse						L{ Rev    f(t) } = 
																			t
Shift							L{ Shift					  f(t) } = Exp[- a s] F(s)
																			a, t

:[font = input; startGroup; ]

LaPlace[ Exp[-Abs[a t]], t, s ]
:[font = print; inactive; startGroup; ]
Rewriting the two-sided expression
 -Abs[t]
E
as a left-sided plus a right-sided function:
 t             CStep[t]
E  CStep[-t] + --------
                   t
                  E
:[font = output; inactive; output; endGroup; endGroup; ]
LTransData[(2*a^2)/(a^2*Abs[a] - s^2*Abs[a]), 
  Rminus[Min[-a, a]], Rplus[Max[-a, a]], LVariables[s]]
;[o]
                      2
                   2 a
LTransData[---------------------, Rminus[Min[-a, a]], 
            2           2
           a  Abs[a] - s  Abs[a]
 
  Rplus[Max[-a, a]], LVariables[s]]
:[font = input; startGroup; ]

LaPlace[ Conjugate[ Exp[a t] CStep[t] ], t, s ]
:[font = output; inactive; output; endGroup; ]
LTransData[Conjugate[-(a + Conjugate[s])^(-1)], 
  Rminus[DirectedInfinity[-1]], Rplus[-Re[a]], LVariables[s]]
;[o]
                              1
LTransData[Conjugate[-(----------------)], 
                       a + Conjugate[s]
 
  Rminus[-Infinity], Rplus[-Re[a]], LVariables[s]]
:[font = input; startGroup; ]

LaPlace[ Convolve[All, All, t]
				 [Exp[a t] CStep[t], Exp[b t] CStep[t]],
		 t, s ]
:[font = output; inactive; output; endGroup; ]
LTransData[1/((-a + s)*(-b + s)), Rminus[Max[Re[a], Re[b]]], 
  Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
                   1
LTransData[-----------------, Rminus[Max[Re[a], Re[b]]], 
           (-a + s) (-b + s)
 
  Rplus[Infinity], LVariables[s]]
:[font = input; startGroup; ]
LaPlace[ Periodic[L, t][ Exp[a t] ] CStep[t], t, s ]
:[font = output; inactive; output; endGroup; ]
LTransData[(E^(L*a) - E^(L*s))/((-1 + E^(L*s))*(a - s)), 
  Rminus[Re[a]], Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
                L a    L s
               E    - E
LTransData[-------------------, Rminus[Re[a]], 
                  L s
           (-1 + E   ) (a - s)
 
  Rplus[Infinity], LVariables[s]]
:[font = input; startGroup; ]
LaPlace[ Rev[t][Exp[a t] CStep[t]], t, s ]
:[font = output; inactive; output; endGroup; ]
LTransData[-(a + s)^(-1), Rminus[Re[a]], 
  Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
               1
LTransData[-(-----), Rminus[Re[a]], Rplus[Infinity], 
             a + s
 
  LVariables[s]]
:[font = input; startGroup; ]

LaPlace[ Shift[L, t][Exp[a t] CStep[t]], t, s ]
:[font = output; inactive; output; endGroup; endGroup; ]
LTransData[1/(E^(L*s)*(-a + s)), Rminus[Re[a]], 
  Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
                 1
LTransData[-------------, Rminus[Re[a]], Rplus[Infinity], 
            L s
           E    (-a + s)
 
  LVariables[s]]
:[font = subsection; inactive; startGroup; Cclosed; ]
Strategies
:[font = text; inactive; ]
		The LaPlace rule base relies on these strategies:
		
(1)	Place affine exponential divided by another affine exponential into one exponential
(2)	Distribute product terms like (t + 1)(t + 2) into t^2 + 3 t + 2
(3) 	Expand out all terms in the numerators of the current expression
(4)	Expand out all terms in the current expression
(5)	Assume that the function is a two-sided function
(6)	Replace all signal processing operators and functions by their mathematical
		definitions
:[font = text; inactive; endGroup; ]
Although similarity is a property, placing the two similarity rules in the property section produced undesirable results, so these rules were moved to this (the last) section.
:[font = subsection; inactive; startGroup; Cclosed; ]
Multidimensional Transforms
:[font = text; inactive; ]

		The LaPlace rule base can find the Laplace transform of multidimensional functions.  Here is the Laplace transform of a two-dimensional separable function:
:[font = input; startGroup; ]

LaPlace[ Exp[a t1] Exp[b t2] CStep[t1,t2], {t1,t2}, {s1,s2} ]
:[font = output; inactive; output; endGroup; ]
LTransData[1/((-a + s1)*(-b + s2)), Rminus[{Re[a], Re[b]}], 
  Rplus[{DirectedInfinity[1], DirectedInfinity[1]}], LVariables[{s1, s2}]]
;[o]
                    1
LTransData[-------------------, Rminus[{Re[a], Re[b]}], 
           (-a + s1) (-b + s2)
 
  Rplus[{Infinity, Infinity}], LVariables[{s1, s2}]]
:[font = text; inactive; ]

Here is a Laplace transform of a non-separable continuous-time function.  This function only takes values along the line t1 = t2.  In this context, Delta[t1 -t2] behaves as a line impulse.  The two-dimensional Laplace transform actually reduces to the one-dimensional transform of Exp[a t] CStep[t] augmented by the substitution s = s1 + s2.  The difficulty here is maintaining the two-dimensional region of convergence:
:[font = input; startGroup; ]

LaPlace[ Exp[a t1] Delta[t1 - t2] CStep[t1,t2], {t1,t2} ]
:[font = output; inactive; output; endGroup; endGroup; ]
LTransData[(-a + s1 + s2)^(-1), Rminus[{Re[a], Re[a]}], 
  Rplus[{DirectedInfinity[1], DirectedInfinity[1]}], 
  LVariables[{s1, s2}]]
;[o]
                1
LTransData[------------, Rminus[{Re[a], Re[a]}], 
           -a + s1 + s2
 
  Rplus[{Infinity, Infinity}], LVariables[{s1, s2}]]
:[font = subsection; inactive; startGroup; Cclosed; ]
Transforms That LaPlace Cannot Find
:[font = text; inactive; ]

		It should be no surprise that the LaPlace rule base cannot compute the Laplace of every function that has one.  This one works now (but it did not use to work):
:[font = input; startGroup; ]

LaPlace[ ( Exp[b t] - Exp[a t] ) CStep[t] / (2 Sqrt[Pi t^3]),
		 t, s ]
:[font = output; inactive; output; endGroup; endGroup; endGroup; ]
LTransData[(-a + s)^(1/2) - (-b + s)^(1/2), 
  Rminus[Max[Re[a], Re[b]]], 
  Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
LTransData[Sqrt[-a + s] - Sqrt[-b + s], 
 
  Rminus[Max[Re[a], Re[b]]], Rplus[Infinity], 
 
  LVariables[s]]
:[font = section; inactive; startGroup; Cclosed; ]
Inverse Laplace Transforms
:[font = text; inactive; startGroup; ]

		The inverse Laplace transform rule base is InvLaPlace.  InvLaPlace returns the two-sided continuous-time function f(t) that represents the Laplace transform F(s): 
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[Oppenheim and Willsky, 1983].  This is an integral of a function of complex variable s (which is different from the forward Laplace transform definition which involves an integral of real variable t).  In this case, the contour of integration is a vertical line through (a, 0) connected with a semicircle of  infinite radius in the s-plane.
:[font = subsection; inactive; startGroup; Cclosed; ]
Invoking the Inverse Laplace Transform Rule Base
:[font = text; inactive; ]
		The calling sequence for InvLaPlace is similar to that for LaPlace:
:[font = input; startGroup; ]

InvLaPlace[ 1/s, s, t ]
:[font = output; inactive; output; endGroup; ]
CStep[t]
;[o]
CStep[t]
:[font = text; inactive; ]

Again, the third argument is optional and defaults to t.  If the time variable is provided, then these options can also be used:
:[font = input; startGroup; ]

Options[ InvLaPlace ]
:[font = output; inactive; output; endGroup; ]
{Apart -> Rational, Dialogue -> True, Simplify -> True, 
  Terms -> 10, TransformLookup -> {}}
;[o]
{Apart -> Rational, Dialogue -> True, Simplify -> True, 
 
  Terms -> 10, TransformLookup -> {}}
:[font = text; inactive; ]

For the meaning of the Dialogue, TransformLookup, and Simplify options, see the Introduction to the Forward Laplace Transforms section.  The Dialogue -> All option will still cause the rule base to display each step of the transformation process. 
:[font = subsubtitle; inactive; startGroup; Cclosed; left; ]
-- Terms Option
:[font = text; inactive; endGroup; ]
		The Terms options specifies how many terms to use if a power series expansion is used.  Setting Terms to False disables the Power Series Strategy for that call to the rule base.  The power series strategy is only used if all other attempts an obtaining the inverse have been tried.
:[font = subsubtitle; inactive; startGroup; Cclosed; left; ]
-- Apart Option
:[font = text; inactive; endGroup; endGroup; ]
		This option is an attempt to work around the way that Mathematica does partial fractions decompositions.  Mathematica requires that the factors be rational numbers.  However, as engineers, there are times that we would like to perform this decomposition with real-valued roots, even though such an expansion violates rigorous mathematics.   When Apart -> Rational, then partial fractions decomposition only occurs when the roots are either rational numbers or symbols.  When Apart -> Real, then partial fractions will be carried out whenever each roots is a floating point number, a rational, or a symbol.  Unfortunately, Apart -> Real means that we have to work around the  Mathematica primitive Apart which results is a very slow partial fractions expansion.
:[font = subsection; inactive; startGroup; Cclosed; ]
Organization of the Rule Base
:[font = text; inactive; ]
		The inverse Laplace transform rule base InvLaPlace has the same overall structure as does the forward Laplace rule base LaPlace, except that the InvLaPlace has no multidimensional hooks since it does not invert transforms representing non-separable multidimensional functions.  Like LaPlace, InvLaPlace works for multidimensional signals as well as two-sided ones.  InvLaPlace calls a one-dimensional inverse Laplace transform rule base once per dimension of the inverse transform:
:[font = output; inactive; locked; output; center; ]
MatrixForm[{"I.   rational transform pairs  24 rules", 
   "II.  non-rat. transform pairs  26 rules", 
   "III. transform properties       8 rules", 
   "IV.  transforms of operators    0 rules", 
   "V.   transform strategies       9 rules"}]
;[o]
I.   rational transform pairs  24 rules
 
II.  non-rat. transform pairs  26 rules
 
III. transform properties       8 rules
 
IV.  transforms of operators    0 rules
 
V.   transform strategies       9 rules
:[font = text; inactive; ]

These one-dimensional Laplace transform rules are maintained in a list instead of being coded directed.  This was necessary because Mathematica does group related rules together--  this feature caused adverse side effects.  So, we made each rule delayed and put them into a list (which is never rearranged by Mathematica).  The length of that list is the number of rules in the one-dimensional Laplace transform rule base:
:[font = input; initialization; startGroup; ]
*)

Length[ SignalProcessing`Analog`InvLaPlace`Private`InvLaPlaceRules ]
(*
:[font = output; inactive; output; endGroup; ]
69
;[o]
69
:[font = text; inactive; ]

		Although this one-dimensional rule base has fewer rules overall than its forward Laplace counterpart, it employs more strategies.  The three new strategies are:
:[font = text; inactive; ]

(1)	normalize the denominator
(2)  approximate the inverse transform using a power series
(3)  partial fraction decomposition
:[font = text; inactive; endGroup; ]

Normalize the denominator means to convert denominators into the form a + b s + c t^2 + ... + d s^(n-1) + s^n. The power series approximation is a Taylor series expansion (in positive powers of s).  The partial fraction decomposition breaks down terms with polynomial denominators into first and second order sections so that they can be more easily inverse transformed.
:[font = subsection; inactive; startGroup; Cclosed; ]
Rational One-Dimensional Transforms
:[font = text; inactive; startGroup; Cclosed; ]

		Inverse transform of a constant:
:[font = input; startGroup; Cclosed; ]

InvLaPlace[ c, s, t ]
:[font = output; inactive; output; endGroup; endGroup; ]
c*Delta[t]
;[o]
c Delta[t]
:[font = text; inactive; startGroup; Cclosed; ]
		The next six examples are inverse transformed by the same transform pair (rule):
:[font = input; startGroup; Cclosed; ]
InvLaPlace[ 1/s, s, t ]
:[font = output; inactive; output; endGroup; ]
CStep[t]
;[o]
CStep[t]
:[font = input; startGroup; Cclosed; ]
InvLaPlace[ 1/s^(3/2), s, t ]
:[font = output; inactive; output; endGroup; ]
(2*t^(1/2)*CStep[t])/Pi^(1/2)
;[o]
2 Sqrt[t] CStep[t]
------------------
     Sqrt[Pi]
:[font = input; startGroup; Cclosed; ]

InvLaPlace[ 1/s^Pi, s, t ]
:[font = output; inactive; output; endGroup; ]
(t^(-1 + Pi)*CStep[t])/Gamma[Pi]
;[o]
 -1 + Pi
t        CStep[t]
-----------------
    Gamma[Pi]
:[font = input; startGroup; Cclosed; ]

InvLaPlace[ 1 / ( s - a ), s, t ]
:[font = output; inactive; output; endGroup; ]
E^(a*t)*CStep[t]
;[o]
 a t
E    CStep[t]
:[font = input; startGroup; Cclosed; ]

InvLaPlace[ 1 / ( s - a )^2, s, t ]
:[font = output; inactive; output; endGroup; ]
E^(a*t)*t*CStep[t]
;[o]
 a t
E    t CStep[t]
:[font = input; startGroup; Cclosed; ]

InvLaPlace[ 1 / ( s - a )^Pi, s, t ]
:[font = output; inactive; output; endGroup; endGroup; ]
(E^(a*t)*t^(-1 + Pi)*CStep[t])/Gamma[Pi]
;[o]
 a t  -1 + Pi
E    t        CStep[t]
----------------------
      Gamma[Pi]
:[font = text; inactive; startGroup; Cclosed; ]
		Multiple poles (handled by partial fractions expansion):
:[font = input; startGroup; Cclosed; ]

InvLaPlace[ 1 / ((s - a)(s - b)), s, t ]
:[font = print; inactive; ]
Performing partial fraction expansion on
        1
-----------------
(-a + s) (-b + s)
in order to break it up into first and
second order sections:
       1                  1
---------------- - ----------------
(a - b) (-a + s)   (a - b) (-b + s)
Taking the inverse Laplace transform ...
:[font = output; inactive; output; endGroup; ]
((E^(a*t) - E^(b*t))*CStep[t])/(a - b)
;[o]
  a t    b t
(E    - E   ) CStep[t]
----------------------
        a - b
:[font = input; startGroup; Cclosed; ]

InvLaPlace[ 1 / ((s - a)(s - b)(s - c)), s, t ]
:[font = print; inactive; ]
Performing partial fraction expansion on
            1
--------------------------
(-a + s) (-b + s) (-c + s)
in order to break it up into first and
second order sections:
                 1                                    1
-(--------------------------------) + ---------------------------------- - 
     2                                           2
  (-a  + a b + a c - b c) (-a + s)    (-(a b) + b  + a c - b c) (-b + s)
 
                  1
  ----------------------------------
                         2
  (-(a b) + a c + b c - c ) (-c + s)
Taking the inverse Laplace transform ...
:[font = output; inactive; output; endGroup; endGroup; ]
(E^(a*t)*CStep[t])/((a - b)*(a - c)) + (E^(b*t)*CStep[t])/((-a + b)*(b - c)) + 
  (E^(c*t)*CStep[t])/((-a + c)*(-b + c))
;[o]
  a t               b t                 c t
 E    CStep[t]     E    CStep[t]       E    CStep[t]
--------------- + ---------------- + -----------------
(a - b) (a - c)   (-a + b) (b - c)   (-a + c) (-b + c)
:[font = text; inactive; startGroup; Cclosed; ]
		Multiple complex-valued poles
:[font = input; startGroup; Cclosed; ]

InvLaPlace[ (s^2 + 3)/(s^2 + 2 s + 2), s, t ]
:[font = print; inactive; startGroup; ]
( After performing partial fraction expansion on
:[font = print; inactive; ]
          2
     3 + s
  ------------
             2
  2 + 2 s + s
:[font = print; inactive; ]
  in order to break it up into first and
:[font = print; inactive; ]
  second order sections:
:[font = print; inactive; ]
        1 - 2 s
  1 + ------------ . )
                 2
      2 + 2 s + s
:[font = output; inactive; output; endGroup; endGroup; ]
(-2*CStep[t]*Cos[t])/E^t + Delta[t] + 
  (3*CStep[t]*Sin[t])/E^t
;[o]
-2 CStep[t] Cos[t]              3 CStep[t] Sin[t]
------------------ + Delta[t] + -----------------
         t                              t
        E                              E
:[font = input; ]
myapart[ fun_, s_ ] :=
	Block [ {newfun, s0},
	         newfun = fun /. s -> s0;
	         rootlist = Sort[ Solve[ newfun == 0, s0 ] ];
	         numroots = Length[rootlist];
	         denom = Product[ s0 /. rootlist[[i]],
	         		          {i, 1, numroots} ];
	         denom ]
:[font = input; startGroup; Cclosed; ]
InvLaPlace[ 1/(s^2 + 2 s + 2)^2, s, t,
			Dialogue -> All ]
:[font = print; inactive; startGroup; ]
  -1             2 -2
L   {(2 + 2 s + s )  }
  s
:[font = print; inactive; ]
which becomes
:[font = print; inactive; ]
  -1             2 -2
L   {(2 + 2 s + s )  }
  s
:[font = print; inactive; ]
which becomes
:[font = message; inactive; ]
Infinity::indt: 
                             -Infinity
   Indeterminate expression s          encountered.
:[font = message; inactive; ]
                                 1
Power::infy: Infinite expression - encountered.
                                 0
:[font = message; inactive; ]
Infinity::indt: 
   Indeterminate expression 0 ComplexInfinity
     encountered.
:[font = print; inactive; ]
  -1
L   {Indeterminate}
  s
:[font = print; inactive; ]
which becomes
:[font = print; inactive; ]
Indeterminate
:[font = print; inactive; ]
which becomes
:[font = print; inactive; ]
Indeterminate
:[font = output; inactive; output; endGroup; endGroup; endGroup; ]
Indeterminate
;[o]
Indeterminate
:[font = text; inactive; startGroup; Cclosed; ]
		Laplace transforms which represent pure sinusoids in the time domain:
:[font = input; startGroup; Cclosed; ]

InvLaPlace[ 1 / ( s^2 + a^2 ), s, t ]
:[font = output; inactive; output; endGroup; ]
(CStep[t]*Sin[a*t])/a
;[o]
CStep[t] Sin[a t]
-----------------
        a
:[font = input; startGroup; Cclosed; ]
InvLaPlace[ s / ( s^2 + a^2 ), s, t ]
:[font = output; inactive; output; endGroup; ]
CStep[t]*Cos[a*t]
;[o]
CStep[t] Cos[a t]
:[font = input; startGroup; Cclosed; ]
InvLaPlace[ 1 / ( s^2 - a^2 ), s, t ]
:[font = print; inactive; ]
Performing partial fraction expansion on
   1
--------
  2    2
-a  + s
in order to break it up into first and
second order sections:
     1              1
------------ - -----------
2 a (-a + s)   2 a (a + s)
Taking the inverse Laplace transform ...
:[font = output; inactive; output; endGroup; ]
(-CStep[t] + E^(2*a*t)*CStep[t])/(2*E^(a*t)*a)
;[o]
             2 a t
-CStep[t] + E      CStep[t]
---------------------------
            a t
         2 E    a
:[font = input; startGroup; Cclosed; ]

InvLaPlace[ s / ( s^2 - a^2 ), s, t ]
:[font = print; inactive; ]
Performing partial fraction expansion on
   s
--------
  2    2
-a  + s
in order to break it up into first and
second order sections:
    1            1
---------- + ---------
2 (-a + s)   2 (a + s)
Taking the inverse Laplace transform ...
:[font = output; inactive; output; endGroup; ]
((1 + E^(2*a*t))*CStep[t])/(2*E^(a*t))
;[o]
      2 a t
(1 + E     ) CStep[t]
---------------------
          a t
       2 E
:[font = input; startGroup; Cclosed; ]

InvLaPlace[ 1 / ( s^2 ( s^2 + a^2 ) ), s, t ]
:[font = print; inactive; ]
Performing partial fraction expansion on
     1
------------
 2   2    2
s  (a  + s )
in order to break it up into first and
second order sections:
  1          1
----- - ------------
 2  2    2   2    2
a  s    a  (a  + s )
Taking the inverse Laplace transform ...
:[font = output; inactive; output; endGroup; ]
(CStep[t]*(a*t - Sin[a*t]))/a^3
;[o]
CStep[t] (a t - Sin[a t])
-------------------------
            3
           a
:[font = input; startGroup; Cclosed; ]

InvLaPlace[ 1 / ( s^2 + a^2 )^2, s, t ]
:[font = output; inactive; output; endGroup; ]
(CStep[t]*(-(a*t*Cos[a*t]) + Sin[a*t]))/(2*a^3)
;[o]
CStep[t] (-(a t Cos[a t]) + Sin[a t])
-------------------------------------
                   3
                2 a
:[font = input; startGroup; Cclosed; ]

InvLaPlace[ s^2 / ( s^2 + a^2 )^2, s, t ]
:[font = print; inactive; ]
Performing partial fraction expansion on
     2
    s
----------
  2    2 2
(a  + s )
in order to break it up into first and
second order sections:
       2
      a            1
-(----------) + -------
    2    2 2     2    2
  (a  + s )     a  + s
Taking the inverse Laplace transform ...
:[font = output; inactive; output; endGroup; ]
(CStep[t]*(a*t*Cos[a*t] + Sin[a*t]))/(2*a)
;[o]
CStep[t] (a t Cos[a t] + Sin[a t])
----------------------------------
               2 a
:[font = input; startGroup; Cclosed; ]

InvLaPlace[ ( s^2 - a^2 ) / ( s^2 + a^2 )^2, s, t ]
:[font = print; inactive; ]
Performing partial fraction expansion on
   2    2
 -a  + s
----------
  2    2 2
(a  + s )
in order to break it up into first and
second order sections:
      2
  -2 a          1
---------- + -------
  2    2 2    2    2
(a  + s )    a  + s
Taking the inverse Laplace transform ...
:[font = output; inactive; output; endGroup; endGroup; ]
t*CStep[t]*Cos[a*t]
;[o]
t CStep[t] Cos[a t]
:[font = text; inactive; startGroup; Cclosed; ]
		Laplace transforms which represent damped sinusoids in the time domain:

:[font = input; startGroup; Cclosed; ]

InvLaPlace[ 1 / ( (s - a)^2 + b^2 ), s, t ]
:[font = print; inactive; ]
Performing partial fraction expansion on
      1
--------------
 2           2
b  + (-a + s)
in order to break it up into first and
second order sections:
         1
--------------------
 2    2            2
a  + b  - 2 a s + s
Taking the inverse Laplace transform ...
:[font = output; inactive; output; endGroup; ]
(E^(a*t)*CStep[t]*Sin[b*t])/b
;[o]
 a t
E    CStep[t] Sin[b t]
----------------------
          b
:[font = input; startGroup; Cclosed; ]
InvLaPlace[ (s - a) / ( (s - a)^2 + b^2 ), s, t ]
:[font = print; inactive; ]
Performing partial fraction expansion on
    -a + s
--------------
 2           2
b  + (-a + s)
in order to break it up into first and
second order sections:
       -a + s
--------------------
 2    2            2
a  + b  - 2 a s + s
Taking the inverse Laplace transform ...
:[font = output; inactive; output; endGroup; ]
E^(a*t)*CStep[t]*Cos[b*t]
;[o]
 a t
E    CStep[t] Cos[b t]
:[font = input; startGroup; Cclosed; ]
InvLaPlace[ 3 a^2 / ( s^3 + a^3 ), s, t ]
:[font = print; inactive; ]
Performing partial fraction expansion on
    2
 3 a
-------
 3    3
a  + s
in order to break it up into first and
second order sections:
  1        2 a - s
----- + -------------
a + s    2          2
        a  - a s + s
Taking the inverse Laplace transform ...
:[font = output; inactive; output; endGroup; ]
CStep[t]/E^(a*t) - E^((a*t)/2)*CStep[t]*Cos[(3^(1/2)*a*t)/2] + 
  (3*E^((a*t)/2)*CStep[t]*Sin[(3^(1/2)*a*t)/2])/3^(1/2)
;[o]
CStep[t]    (a t)/2              Sqrt[3] a t
-------- - E        CStep[t] Cos[-----------] + 
   a t                                2
  E
 
     (a t)/2              Sqrt[3] a t
  3 E        CStep[t] Sin[-----------]
                               2
  ------------------------------------
                Sqrt[3]
:[font = input; startGroup; Cclosed; ]

InvLaPlace[ 4 a^3 / ( s^4 + 4 a^4 ), s, t ]
:[font = print; inactive; ]
Performing partial fraction expansion on
     3
  4 a
---------
   4    4
4 a  + s
in order to break it up into first and
second order sections:
          s                   s
      a - -               a + -
          2                   2
----------------- + -----------------
   2            2      2            2
2 a  - 2 a s + s    2 a  + 2 a s + s
Taking the inverse Laplace transform ...
:[font = output; inactive; output; endGroup; ]
-(CStep[t]*(-Cos[a*t] + E^(2*a*t)*Cos[a*t] - Sin[a*t] - E^(2*a*t)*Sin[a*t]))/
  (2*E^(a*t))
;[o]
                         2 a t                        2 a t
-(CStep[t] (-Cos[a t] + E      Cos[a t] - Sin[a t] - E      Sin[a t]))
----------------------------------------------------------------------
                                   a t
                                2 E
:[font = input; startGroup; Cclosed; ]

InvLaPlace[ s / ( s^4 - a^4 ), s, t ]
:[font = print; inactive; ]
Performing partial fraction expansion on
   s
--------
  4    4
-a  + s
in order to break it up into first and
second order sections:
      1              1               s
------------- + ------------ - --------------
   2               2              2   2    2
4 a  (-a + s)   4 a  (a + s)   2 a  (a  + s )
Taking the inverse Laplace transform ...
:[font = output; inactive; output; endGroup; endGroup; endGroup; ]
(CStep[t]*(1 + E^(2*a*t) - 2*E^(a*t)*Cos[a*t]))/(4*E^(a*t)*a^2)
;[o]
               2 a t      a t
CStep[t] (1 + E      - 2 E    Cos[a t])
---------------------------------------
                  a t  2
               4 E    a
:[font = subsection; inactive; startGroup; Cclosed; ]
Non-rational One-Dimensional Transforms
:[font = subsubsection; inactive; startGroup; Cclosed; ]
Square Root Forms
:[font = text; inactive; startGroup; Cclosed; ]

		Here is a detailed example of an inverse transform involving square root.
:[font = input; ]

InvLaPlace[ (b^2 - a^2) / ((s - a^2)(Sqrt[s] + b)), s, t ]
:[font = print; inactive; ]
Performing partial fraction expansion on
           1
-----------------------
                 2
(b + Sqrt[s]) (-a  + s)
in order to break it up into first and
second order sections:
                1                                 1
-(------------------------------) - ----------------------------- + 
       2                                 2
  (-2 a  - 2 a b) (-a + Sqrt[s])    (-2 a  + 2 a b) (a + Sqrt[s])
 
             1
  ------------------------
     2    2
  (-a  + b ) (b + Sqrt[s])
Taking the inverse Laplace transform ...
:[font = output; inactive; output; ]
CStep[t]*(E^(a^2*t)*b - E^(b^2*t)*b - E^(a^2*t)*a*Erf[a*t^(1/2)] + 
    E^(b^2*t)*b*Erf[b*t^(1/2)])
;[o]
            2         2         2                        2
           a  t      b  t      a  t                     b  t
CStep[t] (E     b - E     b - E     a Erf[a Sqrt[t]] + E     b Erf[b Sqrt[t]])
:[font = text; inactive; ]

Mathematica does not have a complementary error function erfc, where erfc(x) = 1 - erf(x).  The last inverse transform can be rearranged as the following times a step function:
:[font = input; startGroup; ]

Exp[a^2 t] ( b - a Erf[a Sqrt[t]] ) - b Exp[b^2 t] ( 1 - Erf[b Sqrt[t]] )
:[font = output; inactive; output; endGroup; ]
E^(a^2*t)*(b - a*Erf[a*t^(1/2)]) - E^(b^2*t)*b*(1 - Erf[b*t^(1/2)])
;[o]
  2                              2
 a  t                           b  t
E     (b - a Erf[a Sqrt[t]]) - E     b (1 - Erf[b Sqrt[t]])
:[font = text; inactive; ]

We can verify this transform pair by passing it back through the forward Laplace transform:
:[font = input; startGroup; ]

LaPlace[ %23 CStep[t], t, s ]
:[font = output; inactive; output; endGroup; ]
LTransData[(a^2 - b^2)/(a^2*b + a^2*s^(1/2) - b*s - s^(3/2)), 
  Rminus[Max[Re[a^2], Re[b^2]]], Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
                       2    2
                      a  - b                              2       2
LTransData[------------------------------, Rminus[Max[Re[a ], Re[b ]]], 
            2      2                  3/2
           a  b + a  Sqrt[s] - b s - s
 
  Rplus[Infinity], LVariables[s]]
:[font = input; startGroup; ]

Expand[ ( s - a^2 ) ( Sqrt[s] + b ) ]
:[font = output; inactive; output; endGroup; endGroup; ]
-(a^2*b) - a^2*s^(1/2) + b*s + s^(3/2)
;[o]
   2       2                  3/2
-(a  b) - a  Sqrt[s] + b s + s
:[font = text; inactive; startGroup; Cclosed; ]
		The following four inverse transforms are transformed by the same transform pair (the last inverse transform pair gives the general case):
:[font = input; startGroup; ]

InvLaPlace[ 1 / ( Sqrt[s] (s + 4) ), s, t ]
:[font = output; inactive; output; endGroup; ]
(-I/2*CStep[t]*Erf[2*I*t^(1/2)])/E^(4*t)
;[o]
-I
-- CStep[t] Erf[2 I Sqrt[t]]
2
----------------------------
             4 t
            E
:[font = input; startGroup; ]

InvLaPlace[ 1 / ( Sqrt[s] (s - a^2) ), s, t ]
:[font = output; inactive; output; endGroup; ]
(E^(a^2*t)*CStep[t]*Erf[a*t^(1/2)])/a
;[o]
  2
 a  t
E     CStep[t] Erf[a Sqrt[t]]
-----------------------------
              a
:[font = input; startGroup; ]
InvLaPlace[ 1 / ( Sqrt[s] (s + a^2) ), s, t ]
:[font = output; inactive; output; endGroup; ]
(-I*CStep[t]*Erf[I*a*t^(1/2)])/(E^(a^2*t)*a)
;[o]
-I CStep[t] Erf[I a Sqrt[t]]
----------------------------
            2
           a  t
          E     a
:[font = input; startGroup; ]

InvLaPlace[ 1 / ( Sqrt[s + b] (s + a) ), s, t ]
:[font = output; inactive; output; endGroup; endGroup; ]
(E^((-a + b)*t)*CStep[t]*Erf[(-a + b)^(1/2)*t^(1/2)])/(-a + b)^(1/2)
;[o]
 (-a + b) t
E           CStep[t] Erf[Sqrt[-a + b] Sqrt[t]]
----------------------------------------------
                 Sqrt[-a + b]
:[font = text; inactive; startGroup; Cclosed; ]
		The following three inverse transforms are transformed by the same transform pair (the last inverse transform pair gives the general case):
:[font = input; startGroup; ]

InvLaPlace[ Sqrt[s] / ( s + 2 ), s, t ]
:[font = output; inactive; output; endGroup; ]
CStep[t]*(1/(Pi^(1/2)*t^(1/2)) - ((-2)^(1/2)*Erf[(-2)^(1/2)*t^(1/2)])/E^(2*t))
;[o]
                 1           Sqrt[-2] Erf[Sqrt[-2] Sqrt[t]]
CStep[t] (---------------- - ------------------------------)
          Sqrt[Pi] Sqrt[t]                 2 t
                                          E
:[font = input; startGroup; ]

InvLaPlace[ Sqrt[s] / ( s - 4 ), s, t ]
:[font = output; inactive; output; endGroup; ]
CStep[t]*(1/(Pi^(1/2)*t^(1/2)) - 2*E^(4*t)*Erf[2*t^(1/2)])
;[o]
                 1              4 t
CStep[t] (---------------- - 2 E    Erf[2 Sqrt[t]])
          Sqrt[Pi] Sqrt[t]
:[font = input; startGroup; ]

InvLaPlace[ Sqrt[s + b] / (s + c), s, t ]
:[font = output; inactive; output; endGroup; endGroup; ]
CStep[t]*(1/(Pi^(1/2)*t^(1/2)) - 
    E^((b - c)*t)*(b - c)^(1/2)*Erf[(b - c)^(1/2)*t^(1/2)])
;[o]
                 1            (b - c) t
CStep[t] (---------------- - E          Sqrt[b - c] Erf[Sqrt[b - c] Sqrt[t]])
          Sqrt[Pi] Sqrt[t]
:[font = text; inactive; startGroup; Cclosed; ]
		Here are several unrelated examples:

:[font = input; startGroup; ]
InvLaPlace[ 1 / ( Sqrt[s] + a ), s, t ]
:[font = output; inactive; output; endGroup; ]
CStep[t]*(1/(Pi^(1/2)*t^(1/2)) - E^(a^2*t)*a*(1 - Erf[a*t^(1/2)]))
;[o]
                               2
                 1            a  t
CStep[t] (---------------- - E     a (1 - Erf[a Sqrt[t]]))
          Sqrt[Pi] Sqrt[t]
:[font = input; startGroup; ]

InvLaPlace[ 1 / ( Sqrt[s] ( Sqrt[s] + a ) ), s, t ]
:[font = print; inactive; ]
Performing partial fraction expansion on
          1
---------------------
(a + Sqrt[s]) Sqrt[s]
in order to break it up into first and
second order sections:
         1               1
-(---------------) + ---------
  a (a + Sqrt[s])    a Sqrt[s]
Taking the inverse Laplace transform ...
:[font = output; inactive; output; endGroup; ]
-(E^(a^2*t)*CStep[t]*(-1 + Erf[a*t^(1/2)]))
;[o]
    2
   a  t
-(E     CStep[t] (-1 + Erf[a Sqrt[t]]))
:[font = input; startGroup; ]

InvLaPlace[ 1 / ((s + a) Sqrt[s + b]), s, t ]
:[font = output; inactive; output; endGroup; ]
(CStep[t]*Erf[(-a + b)^(1/2)*t^(1/2)])/(E^(a*t)*(-a + b)^(1/2))
;[o]
CStep[t] Erf[Sqrt[-a + b] Sqrt[t]]
----------------------------------
         a t
        E    Sqrt[-a + b]
:[font = input; startGroup; ]

InvLaPlace[ (b^2 - a^2) / ( Sqrt[s] (s - a^2) (Sqrt[s] + b) ), s, t ]
:[font = print; inactive; ]
Performing partial fraction expansion on
               1
-------------------------------
                         2
(b + Sqrt[s]) Sqrt[s] (-a  + s)
in order to break it up into first and
second order sections:
              1                                1
------------------------------ - ----------------------------- + 
    3      2                         3      2
(2 a  + 2 a  b) (-a + Sqrt[s])   (2 a  - 2 a  b) (a + Sqrt[s])
 
              1                    1
  ------------------------- - ------------
    2      3                   2
  (a  b - b ) (b + Sqrt[s])   a  b Sqrt[s]
Taking the inverse Laplace transform ...
:[font = output; inactive; output; endGroup; ]
-((CStep[t]*(E^(a^2*t)*a - E^(b^2*t)*a - E^(a^2*t)*b*Erf[a*t^(1/2)] + 
        E^(b^2*t)*a*Erf[b*t^(1/2)]))/a)
;[o]
              2         2         2                        2
             a  t      b  t      a  t                     b  t
  CStep[t] (E     a - E     a - E     b Erf[a Sqrt[t]] + E     a Erf[b Sqrt[t]])
-(------------------------------------------------------------------------------)
                                        a
:[font = input; startGroup; ]

InvLaPlace[ 1 / ( Sqrt[(s + a)(s + b)] ), s, t ]
:[font = output; inactive; output; endGroup; ]
(BesselI[0, ((a - b)*t)/2]*CStep[t])/E^(((a + b)*t)/2)
;[o]
           (a - b) t
BesselI[0, ---------] CStep[t]
               2
------------------------------
         ((a + b) t)/2
        E
:[font = input; startGroup; ]

InvLaPlace[ ( Sqrt[s^2 + a^2] - s )^2 / Sqrt[s^2 + a^2], s, t ]
:[font = output; inactive; output; endGroup; ]
a^2*BesselJ[2, a*t]*CStep[t]
;[o]
 2
a  BesselJ[2, a t] CStep[t]
:[font = input; startGroup; ]

InvLaPlace[ ( s - Sqrt[s^2 - a^2] )^Pi / Sqrt[s^2 - a^2], s, t ]
:[font = output; inactive; output; endGroup; ]
a^Pi*BesselI[Pi, a*t]*CStep[t]
;[o]
 Pi
a   BesselI[Pi, a t] CStep[t]
:[font = input; startGroup; ]

InvLaPlace[ 1 / Sqrt[ s^2 + a^2 ], s, t ]
:[font = output; inactive; output; endGroup; endGroup; endGroup; endGroup; ]
BesselJ[0, a*t]*CStep[t]
;[o]
BesselJ[0, a t] CStep[t]
:[font = subsection; inactive; startGroup; Cclosed; ]
Properties
:[font = text; inactive; ]

		The LaPlace rule base implements the following properties:

Additivity																							Linv{A(s) + B(s)}			 = Linv{A(s)} + Linv{B(s)}
Homogeneity																						Linv{c A(s)}														= c L{A(s)}
Multiplication-by-Exponential						Linv{exp(c s) F(s)}		= Linv{F(s)}, t -> t + c
Similarity																							Linv{F(c s}}														= Linv{F(s)} / |c|, s -> s/c
Pole-in-Denominator																			Linv{F(s)/(s + a)}				= exp(- a t) Linv{F(s - a) / s}
Multiplication-by-Frequency									Linv{s F(s)}														= df(t)/dt + f(0+)
																																																																								t																																																																					
Division-by-Frequency																		Linv{F(s)/s}															=  	/  f(r) dr
																																																																					0
:[font = subsubsection; inactive; startGroup; Cclosed; ]
Additivity
:[font = text; inactive; ]

		The Laplace transform of A(s) + B(s) is a(t), the inverse Laplace transform of A(s), plus b(t), the inverse Laplace transform of B(s):
:[font = input; startGroup; ]

InvLaPlace[ 1 + Exp[-s] / s, s, t, Dialogue -> All ]
:[font = print; inactive; ]
  -1      1
L   {1 + ----}
  s       s
         E  s
becomes
  -1        -1  1
L   {1} + L   {----}
  s         s   s
               E  s
becomes
              -1 1
Delta[t] + {L   {-}}
              s  s  t -> -1 + t
becomes
Delta[t] + {CStep[t]}
                     t -> -1 + t
becomes
CStep[-1 + t] + Delta[t]
becomes
CStep[-1 + t] + Delta[t]
:[font = output; inactive; output; endGroup; endGroup; ]
CStep[-1 + t] + Delta[t]
;[o]
CStep[-1 + t] + Delta[t]
:[font = subsubsection; inactive; startGroup; Cclosed; ]
Homogeneity
:[font = text; inactive; ]

		This property moves all terms not dependent on the time variable outside of the Laplace transform operator with no effect on the region of convergence:

:[font = input; startGroup; ]
InvLaPlace[ A K / (2 s), s, t ]
:[font = output; inactive; output; endGroup; endGroup; ]
(A*K*CStep[t])/2
;[o]
A K CStep[t]
------------
     2
:[font = subsubsection; inactive; startGroup; Cclosed; ]
Multiplication-by-Exponential
:[font = text; inactive; ]

		The multiplication of f(t) by exp(a t)  shifts F(s), the Laplace transform of  f(t), to the right by a, and the region of convergence is shifted by the real part of a:
:[font = input; startGroup; ]

InvLaPlace[ s / ( 1 + s^2 ), s, t ]
:[font = output; inactive; output; endGroup; ]
CStep[t]*Cos[t]
;[o]
CStep[t] Cos[t]
:[font = input; startGroup; ]

InvLaPlace[ Exp[-a s] s / ( 1 + s^2 ), s, t ]
:[font = output; inactive; output; endGroup; endGroup; ]
CStep[-a + t]*Cos[-a + t]
;[o]
CStep[-a + t] Cos[-a + t]
:[font = subsubsection; inactive; startGroup; Cclosed; ]
Similarity
:[font = input; startGroup; ]

InvLaPlace[ 1 / ((s + a) Sqrt[s + b]), s, t ]
:[font = output; inactive; output; endGroup; ]
(CStep[t]*Erf[(-a + b)^(1/2)*t^(1/2)])/(E^(a*t)*(-a + b)^(1/2))
;[o]
CStep[t] Erf[Sqrt[-a + b] Sqrt[t]]
----------------------------------
         a t
        E    Sqrt[-a + b]
:[font = input; startGroup; ]

InvLaPlace[ 1 / ((c s + a) Sqrt[c s + b]), s, t ]
:[font = output; inactive; output; endGroup; endGroup; ]
(CStep[t/c]*Erf[((-a + b)^(1/2)*t^(1/2))/c^(1/2)])/
  (E^((a*t)/c)*(-a + b)^(1/2)*Abs[c])
;[o]
      t      Sqrt[-a + b] Sqrt[t]
CStep[-] Erf[--------------------]
      c            Sqrt[c]
----------------------------------
    (a t)/c
   E        Sqrt[-a + b] Abs[c]
:[font = subsubsection; inactive; startGroup; Cclosed; ]
Pole-in-Denominator
:[font = input; startGroup; ]

InvLaPlace[ 1 / ( s - 1 ), s, t ]
:[font = output; inactive; output; endGroup; ]
E^t*CStep[t]
;[o]
 t
E  CStep[t]
:[font = input; startGroup; ]
InvLaPlace[ Log[1 + s], s, t ]
:[font = output; inactive; output; endGroup; ]
-(CStep[t]/(E^t*t))
;[o]
  CStep[t]
-(--------)
     t
    E  t
:[font = input; startGroup; ]

InvLaPlace[ Log[1 + s] / (( s + a )(s + b)), s, t ]
:[font = print; inactive; ]
Performing partial fraction expansion on
  Log[1 + s]
---------------
(a + s) (b + s)
in order to break it up into first and
second order sections:
   Log[1 + s]         Log[1 + s]
---------------- - ----------------
(-a + b) (a + s)   (-a + b) (b + s)
Taking the inverse Laplace transform ...
:[font = output; inactive; output; endGroup; endGroup; ]
(CStep[t]*(-ExpIntegralEi[(-1 + a)*t] + Log[1 - a]))/(E^(a*t)*(-a + b)) - 
  (CStep[t]*(-ExpIntegralEi[(-1 + b)*t] + Log[1 - b]))/(E^(b*t)*(-a + b))
;[o]
CStep[t] (-ExpIntegralEi[(-1 + a) t] + Log[1 - a])
-------------------------------------------------- - 
                   a t
                  E    (-a + b)
 
  CStep[t] (-ExpIntegralEi[(-1 + b) t] + Log[1 - b])
  --------------------------------------------------
                     b t
                    E    (-a + b)
:[font = subsubsection; inactive; startGroup; Cclosed; ]
Multiplication-by-Frequency
:[font = text; inactive; ]

		Multiplication in frequency corresponds to differentiation in the time domain:
:[font = input; startGroup; ]

InvLaPlace[ 1 / Sqrt[(s + a)(s - a)], s, t ]
:[font = output; inactive; output; endGroup; ]
BesselI[0, -(a*t)]*CStep[t]
;[o]
BesselI[0, -(a t)] CStep[t]
:[font = input; startGroup; ]

InvLaPlace[ s / Sqrt[(s + a)(s - a)], s, t ]
:[font = output; inactive; output; endGroup; ]
-(a*(BesselI[-1, -(a*t)] + BesselI[1, -(a*t)])*CStep[t])/2 + Delta[t]
;[o]
-(a (BesselI[-1, -(a t)] + BesselI[1, -(a t)]) CStep[t])
-------------------------------------------------------- + Delta[t]
                           2
:[font = input; startGroup; ]

LaPlace[ -a BesselI[1, - a t] CStep[t] + Delta[t], t, s ]
:[font = output; inactive; output; endGroup; endGroup; ]
LTransData[s/(-a^2 + s^2)^(1/2), Rminus[Abs[Re[-a]]], 
  Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
                 s
LTransData[--------------, Rminus[Abs[Re[-a]]], Rplus[Infinity], LVariables[s]]
                  2    2
           Sqrt[-a  + s ]
:[font = subsubsection; inactive; startGroup; Cclosed; ]
Division-by-Frequency
:[font = text; inactive; ]

		Multiplication in frequency corresponds to differentiation in the time domain:
:[font = input; startGroup; ]

InvLaPlace[ 1 / Sqrt[(s + a)(s - a)], s, t ]
:[font = output; inactive; output; endGroup; ]
BesselI[0, -(a*t)]*CStep[t]
;[o]
BesselI[0, -(a t)] CStep[t]
:[font = input; startGroup; ]

InvLaPlace[ 1 / (s Sqrt[(s + a)(s - a)]), s, t ]
:[font = output; inactive; output; endGroup; ]
CStep[t]*Integrate[BesselI[0, -(a*tau3)], {tau3, 0, t}]
;[o]
CStep[t] Integrate[BesselI[0, -(a tau3)], {tau3, 0, t}]
:[font = text; inactive; endGroup; endGroup; ]

This is the integration of the Bessel function Io(- a t) with respect to t. 
:[font = subsection; inactive; startGroup; Cclosed; ]
Strategies
:[font = text; inactive; ]

		The rule base employs the following strategies:
:[font = text; inactive; ]

Partial Fractions
Normalize the Denominator
Factor the Denominator
Distribute
Expand Numerators
Expand All Terms
Assume a Two-Sided Transform
Replace Signal Processing Primitives with Mathematica Ones
Power Series Expansion

:[font = subsubsection; inactive; startGroup; Cclosed; ]
Partial Fractions Decomposition
:[font = text; inactive; ]
		The key strategy is partial fractions decomposition which can be used for more than just s-domain polynomials.  They are also useful in transforming F(s)/s forms as well as G(s)/((s + a)(s + b)) forms, where F(s) and G(s) are not necessarily rational polynomials in s.  Partial fractions is handled by the Mathematica primitive Apart.  It is encoded as three rules--  one for rational polynomials, one for denominator poles, and one in this section of the rule base.
:[font = text; inactive; ]
		The philosophy underlying Apart is to perform partial fractions on a ratio of two rational polynomials, so it does not handle real-valued coefficients (so we have added the Apart option to InvLaPlace).  In Mathematica 1.2, the Apart function is not robust because it does not handle repeated complex-valued roots in the denominator.  For example,
:[font = input; startGroup; ]
Apart[ (s^2 - 1) / ( s^2 + 2 s + 2 )^2, s ]
:[font = output; inactive; output; endGroup; ]
(-3 - 2*s)/(2 + 2*s + s^2)^2 + (2 + 2*s + s^2)^(-1)
;[o]
   -3 - 2 s            1
--------------- + ------------
            2 2              2
(2 + 2 s + s )    2 + 2 s + s
:[font = text; inactive; ]
But this is really not the complete partial fractions decomposition.  Rather, this is:
:[font = input; startGroup; ]
MyApart[ (s^2 - 1) / ( s^2 + 2 s + 2 )^2, s ]
:[font = output; inactive; output; endGroup; ]
(1/4 + I/2)/(1 - I + s)^2 + -I/4/(1 - I + s) + 
  (1/4 - I/2)/(1 + I + s)^2 + I/4/(1 + I + s)
;[o]
   1   I          -I          1   I           I
   - + -          --          - - -           -
   4   2          4           4   2           4
------------ + --------- + ------------ + ---------
           2   1 - I + s              2   1 + I + s
(1 - I + s)                (1 + I + s)
:[font = text; inactive; ]
		Now, let's try to compute the inverse Laplace transform:
:[font = input; startGroup; Cclosed; ]
InvLaPlace[ (s^2 - 1) / ( s^2 + 2 s + 2 )^2, s, t ]
:[font = print; inactive; startGroup; ]
( After performing partial fraction expansion on
:[font = print; inactive; ]
            2
      -1 + s
  ---------------
              2 2
  (2 + 2 s + s )
:[font = print; inactive; ]
  into its exact roots:
:[font = print; inactive; ]
     -3 - 2 s            1
  --------------- + ------------ . )
              2 2              2
  (2 + 2 s + s )    2 + 2 s + s
:[font = print; inactive; ]
( After performing partial fraction expansion on
:[font = print; inactive; ]
     -3 - 2 s
  ---------------
              2 2
  (2 + 2 s + s )
:[font = print; inactive; ]
  into its exact roots:
:[font = print; inactive; ]
     1   I           I          1   I          -I
     - + -           -          - - -          --
     4   2           4          4   2          4
  ------------ + --------- + ------------ + ---------
             2   1 - I + s              2   1 + I + s
  (1 - I + s)                (1 + I + s)
 
   . )
:[font = output; inactive; output; endGroup; endGroup; endGroup; ]
(E^((-1 - I)*t)*(-I*CStep[t] + I*E^(2*I*t)*CStep[t] + 
      (1 - 2*I)*t*CStep[t] + 
      (1 + 2*I)*E^(2*I*t)*t*CStep[t] + 
      4*E^(I*t)*CStep[t]*Sin[t]))/4
;[o]
  (-1 - I) t                   2 I t
(E           (-I CStep[t] + I E      CStep[t] + 
 
      (1 - 2 I) t CStep[t] + 
 
                 2 I t
      (1 + 2 I) E      t CStep[t] + 
 
         I t
      4 E    CStep[t] Sin[t])) / 4
:[font = subsubsection; inactive; startGroup; Cclosed; ]
Power Series Expansion
:[font = text; inactive; endGroup; endGroup; ]
		One strategy-- power series expansion-- is only applied when all other attempts at the inverse transform have failed.  In general, the expansion is a Taylor series expansion about s = 0 (positive powers of s).  The number of terms in the expansion is controlled by the Terms options (see the Introduction to this section).
:[font = subsection; inactive; startGroup; Cclosed; ]
Multidimensional Transforms
:[font = text; inactive; ]

		The multidimensional inverse Laplace transform only works for separable functions.
:[font = input; startGroup; ]

InvLaPlace[ 1 / ((s1 + a)(s2 + b)), {s1, s2} ]
:[font = output; inactive; output; endGroup; ]
E^(-(a*t1) - b*t2)*CStep[t1]*CStep[t2]
;[o]
 -(a t1) - b t2
E               CStep[t1] CStep[t2]
:[font = input; startGroup; ]
InvLaPlace[ 1 / Sqrt[(s1 + a)(s2 + b)], {s1, s2}, {u1, u2} ]
:[font = output; inactive; output; endGroup; endGroup; endGroup; ]
(E^(-(a*u1) - b*u2)*CStep[u1]*CStep[u2])/(Pi*u1^(1/2)*u2^(1/2))
;[o]
 -(a u1) - b u2
E               CStep[u1] CStep[u2]
-----------------------------------
       Pi Sqrt[u1] Sqrt[u2]
:[font = section; inactive; startGroup; Cclosed; ]
Solving Differential Equations
:[font = text; inactive; ]

		The Mathematica object LSolve uses the Laplace transform rule bases to solve linear constant coefficient differential equations.   LSolve introduces the partial Laplace transform which expresses the Laplace transform of y'(t) as  s Y(s) - y(0)  when y(t) is unknown.
:[font = subsection; inactive; startGroup; Cclosed; ]
Differential Equation With Zero-Valued Initial Conditions
:[font = text; inactive; ]

		The first example is a second-order linear constant-coefficient differential equation whose initial conditions are zero:  y'(0) = y(0) = 0.  The Mathematica object LSolve requires two arguments:  the differential equation to be solved and what to solve for.
:[font = input; startGroup; ]

LSolve[ y''[t] + 3/2 y'[t] + 1/2 y[t] == Exp[a t] CStep[t], y[t] ]
:[font = output; inactive; output; endGroup; ]
(4*CStep[t/2])/(E^(t/2)*(-1 - 2*a)) - (2*CStep[t])/(E^t*(-1 - a)) - 
  (2*E^(a*t)*CStep[t])/(-1 - 3*a - 2*a^2)
;[o]
          t
  4 CStep[-]                       a t
          2       2 CStep[t]    2 E    CStep[t]
--------------- - ----------- - ---------------
 t/2               t                          2
E    (-1 - 2 a)   E  (-1 - a)   -1 - 3 a - 2 a
:[font = text; inactive; ]

Implied here is that we are looking for the solution to the differential equation when t > 0.  Therefore, we can drop the continuous-time step functions since they are 1 for t > 0.
:[font = subsubsection; inactive; startGroup; Cclosed; ]
Verifying a Solution
:[font = text; inactive; ]

We can now use Mathematica to verify the solution, which we assign to the function ysol:
:[font = input; initialization; startGroup; ]
*)

ysol =	4 Exp[-t/2] / (-1 - 2 a) -
		2 Exp[-t] / (-1 - a) -
		2 Exp[a t] / (-1 - 3 a - 2 a^2 )
(*
:[font = output; inactive; output; endGroup; ]
4/(E^(t/2)*(-1 - 2*a)) - 2/(E^t*(-1 - a)) - 
  (2*E^(a*t))/(-1 - 3*a - 2*a^2)
;[o]
                                       a t
       4               2            2 E
--------------- - ----------- - ---------------
 t/2               t                          2
E    (-1 - 2 a)   E  (-1 - a)   -1 - 3 a - 2 a
:[font = text; inactive; ]

 We can now check to make sure that the initial conditions are valid.  First, we check to see if y(0+) = 0:
:[font = input; startGroup; ]
ysol /. t -> 0

:[font = output; inactive; output; endGroup; ]
4/(-1 - 2*a) - 2/(-1 - a) - 2/(-1 - 3*a - 2*a^2)
;[o]
   4         2             2
-------- - ------ - ---------------
-1 - 2 a   -1 - a                 2
                    -1 - 3 a - 2 a
:[font = text; inactive; ]

This may not look like zero but it actually is:

:[font = input; startGroup; ]

Simplify[ ysol /. t -> 0 ]
:[font = output; inactive; output; endGroup; ]
0
;[o]
0
:[font = text; inactive; ]

Second, we can check to make sure that y'(0+) equals zero:
:[font = input; initialization; startGroup; ]
*)

Simplify[ D[ ysol, t ] /. t -> 0 ]
(*
:[font = output; inactive; output; endGroup; ]
0
;[o]
0
:[font = text; inactive; ]

Next, we can try to verify that the solution satisfies the differential equation:
:[font = input; startGroup; ]

ysolprime = D[ ysol, t];
ysoldblprime = D[ ysolprime, t];

Simplify[ ysoldblprime + 3/2 ysolprime + 1/2 ysol ]
:[font = output; inactive; output; endGroup; endGroup; ]
E^(a*t)
;[o]
 a t
E
:[font = subsubsection; inactive; startGroup; Cclosed; ]
Analyzing a Solution
:[font = text; inactive; ]

		Now that we shown that the solution is correct, we can analyze the solution.  First, there are certain values of the free parameter a which yield an unbounded (and hence invalid) solution.  This occurs when any of the denominator terms become zero, which happens when a = -1/2 and when a = -1.  This can be seen from inspection, but Mathematica can determine these values in the general case.  First, we convert the solution from its form of Plus[term1, term2, term3] to a list of the form List[term1, term2, term3] by using Apply and then solve the cases where each denominator is zero using Solve:
:[font = input; startGroup; ]

Map[ Solve[Denominator[#1] == 0, a]&, Apply[List, ysol] ]
:[font = output; inactive; output; endGroup; endGroup; ]
{{{a -> -1/2}}, {{a -> -1}}, {{a -> -1}, {a -> -1/2}}}
;[o]
          1                                      1
{{{a -> -(-)}}, {{a -> -1}}, {{a -> -1}, {a -> -(-)}}}
          2                                      2
:[font = subsubsection; inactive; startGroup; Cclosed; ]
Interpreting a Solution
:[font = text; inactive; ]

		In this case, ysol is assigned to the solution of the differential equation automatically by the notebook.  To plot the solution, then, we simplify plot ysol for different values of a:
:[font = input; startGroup; ]

Plot3D[ ysol, {t, 0, 1}, {a, 0, 1}, AxesLabel -> { "t", "a", " " } ]
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The Unformatted text for this cell was not generated.
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;[o]
-SurfaceGraphics-
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Alternatively, we could set a to a constant value and plot y(t).  For a = -2, y(t) is plotted below:
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Plot[ Release[ysol /. a -> -2], {t, 0, 2}, AxesLabel -> { "t", "y(t)" } ]
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%!
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The Unformatted text for this cell was not generated.
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when Unformatted text is generated.
;[o]
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Differential Equation With Initial Conditions
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		The analysis carried out for a differential equation with zero-valued initial conditions applies to those with non-zero initial conditions.  Here is the same differential equation with y'(0) = 1/2 and y(0) = 4:
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LSolve[ y''[t] + 3/2 y'[t] + 1/2 y[t] == Exp[a t] CStep[t],
		y[t], y'[0] -> 1/2, y[0] -> 4 ]
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((-10 - 36*a)*CStep[t/2])/(2*E^(t/2)*(-1 - 2*a)) + 
  ((3 + 5*a)*CStep[t])/(E^t*(-1 - a)) - (2*E^(a*t)*CStep[t])/(-1 - 3*a - 2*a^2)
;[o]
                   t
(-10 - 36 a) CStep[-]                           a t
                   2    (3 + 5 a) CStep[t]   2 E    CStep[t]
--------------------- + ------------------ - ---------------
     t/2                    t                              2
  2 E    (-1 - 2 a)        E  (-1 - a)       -1 - 3 a - 2 a
:[font = text; inactive; ]

Again, implied here is that we want the solution to the differential equation when t > 0.  So once again, we will drop the step functions since they are 1 for t > 0.
:[font = subsubsection; inactive; startGroup; Cclosed; ]
Verifying a Solution
:[font = text; inactive; ]

We can now use Mathematica to verify the solution, which we assign to the function ysol:
:[font = input; initialization; startGroup; ]
*)

yicsol =	(5 + 18 a) Exp[-t/2] / (1 + 2 a) +
			(3 + 5 a) Exp[-t] / (-1 - a) -
			2 Exp[a t] / (-1 - 3 a - 2 a^2 )
(*
:[font = output; inactive; output; endGroup; ]
(3 + 5*a)/(E^t*(-1 - a)) + 
  (5 + 18*a)/(E^(t/2)*(1 + 2*a)) - 
  (2*E^(a*t))/(-1 - 3*a - 2*a^2)
;[o]
                                      a t
  3 + 5 a        5 + 18 a          2 E
----------- + -------------- - ---------------
 t             t/2                           2
E  (-1 - a)   E    (1 + 2 a)   -1 - 3 a - 2 a
:[font = text; inactive; ]

 We can now check to make sure that the initial conditions are valid.  First, we check to see if y(0+) = 4:
:[font = input; startGroup; ]
yicsol /. t -> 0

:[font = output; inactive; output; endGroup; ]
(3 + 5*a)/(-1 - a) + (5 + 18*a)/(1 + 2*a) - 2/(-1 - 3*a - 2*a^2)
;[o]
3 + 5 a   5 + 18 a          2
------- + -------- - ---------------
-1 - a    1 + 2 a                  2
                     -1 - 3 a - 2 a
:[font = text; inactive; ]

This may not look like four but it actually is:

:[font = input; startGroup; ]

Simplify[ yicsol /. t -> 0 ]
:[font = output; inactive; output; endGroup; ]
4
;[o]
4
:[font = text; inactive; ]

Second, we can check to make sure that ysol'(0+) equals 1/2:
:[font = input; initialization; startGroup; ]
*)

Simplify[ D[ yicsol, t ] /. t -> 0 ]
(*
:[font = output; inactive; output; endGroup; ]
1/2
;[o]
1
-
2
:[font = text; inactive; ]

Next, we can try to verify that the solution satisfies the differential equation:
:[font = input; startGroup; ]

yicsolprime = D[ yicsol, t];
yicsoldblprime = D[ yicsolprime, t];

Simplify[ yicsoldblprime + 3/2 yicsolprime + 1/2 yicsol ]
:[font = output; inactive; output; endGroup; endGroup; ]
E^(a*t)
;[o]
 a t
E
:[font = subsubsection; inactive; startGroup; Cclosed; ]
Analyzing a Solution
:[font = text; inactive; ]

		Now that we shown that the solution is correct, we can analyze the solution.  First, there are certain values of the free parameter a which yield an unbounded (and hence invalid) solution.  This occurs when any of the denominator terms become zero, which happens when a = -1/2 and when a = -1.  This can be seen from inspection, but Mathematica can determine these values in the general case. Again, we convert the solution from its form of Plus[term1, term2, term3] to a list of the form List[term1, term2, term3] by using Apply and then solve the cases where each denominator is zero using Solve:
:[font = input; startGroup; ]

Map[ Solve[Denominator[#1] == 0, a]&, Apply[List, yicsol] ]
:[font = output; inactive; output; endGroup; endGroup; ]
{{{a -> -1}}, {{a -> -1/2}}, {{a -> -1}, {a -> -1/2}}}
;[o]
                       1                         1
{{{a -> -1}}, {{a -> -(-)}}, {{a -> -1}, {a -> -(-)}}}
                       2                         2
:[font = subsubsection; inactive; startGroup; Cclosed; ]
Interpreting a Solution
:[font = text; inactive; ]

		In this case, ysol is assigned to the solution of the differential equation automatically by the notebook.  To plot the solution, then, we simplify plot ysol for different values of a:
:[font = input; startGroup; ]

Plot3D[ yicsol, {t, 0, 1}, {a, 0, 1}, AxesLabel -> { "t", "a", " " } ]
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MathPictureEnd
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The Unformatted text for this cell was not generated.
Use options in the Actions Settings dialog box to control
when Unformatted text is generated.
;[o]
-SurfaceGraphics-
:[font = text; inactive; ]

Alternatively, we could set a to a constant value and plot y(t).  For a = -2, y(t) is plotted below:
:[font = input; startGroup; ]

Plot[ Release[yicsol /. a -> -2], {t, 0, 2}, AxesLabel -> { "t", "y(t)" } ]
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%!
%%Creator: Mathematica
%%AspectRatio: 0.61803 
MathPictureStart
% Scaling calculations
0.02381 0.47619 -1.38206 0.48731 [
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[(1.5)] 0.7381 0.06738 0 1 Msboxa
[(2)] 0.97619 0.06738 0 1 Msboxa
[(t)] 1.00625 0.07988 -1 0 Msboxa
[(3.2)] 0.01131 0.17734 1 0 Msboxa
[(3.4)] 0.01131 0.2748 1 0 Msboxa
[(3.6)] 0.01131 0.37226 1 0 Msboxa
[(3.8)] 0.01131 0.46972 1 0 Msboxa
[(4)] 0.01131 0.56719 1 0 Msboxa
[(y\(t\))] 0.02381 0.62428 0 -1 Msboxa
[ -0.001 -0.001 0 0 ]
[ 1.001 0.61903 0 0 ]
] MathScale
% Start of Graphics
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Mfstroke
grestore
grestore
% End of Graphics
MathPictureEnd
:[font = output; inactive; output; endGroup; endGroup; endGroup; endGroup; ]
The Unformatted text for this cell was not generated.
Use options in the Actions Settings dialog box to control
when Unformatted text is generated.
;[o]
-Graphics-
:[font = section; inactive; startGroup; Cclosed; ]
Analysis
:[font = subsection; inactive; startGroup; Cclosed; ]
Stability Analysis
:[font = text; inactive; ]
		Since the Laplace transform tracks the region of convergence, the object Stable can often tell if a signal is stable if its Laplace transform's region of convergence is a subset of (0, Infinity).
:[font = input; startGroup; ]
LaPlace[ Exp[a t] CStep[t], t]
:[font = output; inactive; output; endGroup; ]
LTransData[(-a + s)^(-1), Rminus[Re[a]], 
  Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
             1
LTransData[------, Rminus[Re[a]], Rplus[Infinity], 
           -a + s
 
  LVariables[s]]
:[font = input; startGroup; ]
Stable[%]
:[font = output; inactive; output; endGroup; ]
Re[a] < 0
;[o]
Re[a] < 0
:[font = text; inactive; ]
This, of course, extends to multiple dimensions:
:[font = input; startGroup; ]
LaPlace[ Exp[a t1] Exp[b t2] CStep[t1, t2], {t1, t2} ]
:[font = output; inactive; output; endGroup; ]
LTransData[1/((-a + s1)*(-b + s2)), Rminus[{Re[a], Re[b]}], 
  Rplus[{DirectedInfinity[1], DirectedInfinity[1]}], 
  LVariables[{s1, s2}]]
;[o]
                    1
LTransData[-------------------, Rminus[{Re[a], Re[b]}], 
           (-a + s1) (-b + s2)
 
  Rplus[{Infinity, Infinity}], LVariables[{s1, s2}]]
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Stable[%]
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Re[a] < 0 && Re[b] < 0
;[o]
Re[a] < 0 && Re[b] < 0
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General Analysis
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		The analog signal processing packages provide ASPAnalyze which is a high level analysis tool.  It plots time-domain expressions and displays the poles and zeros of the expression's transform.  It checks stability of the signal.  It plots the magnitude and frequency responses.
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Example 1
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ASPAnalyze[ Delta[t + 1] + Delta[t - 1], t, -2, 2 ]
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Delta[-1 + t] + Delta[1 + t]
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has the following Laplace transform:
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 -s    s
E   + E
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The strip of convergence is:
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-Infinity < Re(s) < Infinity
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The system is stable.
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PoleZeroPlot::notrational: 
   Transform is not a rational polynomial.
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Solve::ifun: 
   Warning: inverse functions are being used by Solve, so
    some solutions may not be found.
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Solve::ifun: 
   Warning: inverse functions are being used by Solve, so
    some solutions may not be found.
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The zeroes are:  {1.5708 I, -1.5708 I}
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The poles are:   {}
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Delta[-1 + t] + Delta[1 + t]
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has the following frequency response:
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 -I w    I w
E     + E
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ASPAnalyze::notinteresting: 
   Could not determine the important section of the
    frequency response.
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LTransData[E^(-s) + E^s, Rminus[DirectedInfinity[-1]], 
  Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
            -s    s
LTransData[E   + E , Rminus[-Infinity], Rplus[Infinity], 
 
  LVariables[s]]
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Example 2
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ASPAnalyze[ t Exp[- a t] Cos[3 Pi t / 16] CStep[t], t ]
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For plotting only, these symbols will
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               3 Pi t
t CStep[t] Cos[------]
                 16
----------------------
          a t
         E
:[font = print; inactive; ]
has the following Laplace transform:
:[font = print; inactive; ]
                2
      -2 (a + s)               1
-(------------------- + ----------------)
       2                    2
   9 Pi           2 2   9 Pi           2
  (----- + (a + s) )    ----- + (a + s)
    256                  256
:[font = print; inactive; ]
The strip of convergence is:
:[font = print; inactive; ]
-Re[a] < Re(s) < Infinity
:[font = print; inactive; ]
The system is stable if Re[a] > 0
:[font = print; inactive; ]

:[font = print; inactive; ]
The zeroes are:  {-1.58905, -0.410951}
:[font = print; inactive; ]
The poles are:   {-1. + 0.589049 I, -1. - 0.589049 I, 
 
   -1. + 0.589049 I, -1. - 0.589049 I}
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               3 Pi t
t CStep[t] Cos[------]
                 16
----------------------
          a t
         E
:[font = print; inactive; ]
has the following frequency response:
:[font = print; inactive; ]
                                      2
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-(------------------ - ---------------------)
      2                     2
  9 Pi             2    9 Pi             2 2
  ----- + (a + I w)    (----- + (a + I w) )
   256                   256
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LTransData[-((-2*(a + s)^2)/
      ((9*Pi^2)/256 + (a + s)^2)^2 + 
     ((9*Pi^2)/256 + (a + s)^2)^(-1)), Rminus[-Re[a]], 
  Rplus[DirectedInfinity[1]], LVariables[s]]
;[o]
                           2
                 -2 (a + s)               1
LTransData[-(------------------- + ----------------), 
                  2                    2
              9 Pi           2 2   9 Pi           2
             (----- + (a + s) )    ----- + (a + s)
               256                  256
 
  Rminus[-Re[a]], Rplus[Infinity], LVariables[s]]
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This work was supported in part by the Joint Services Electronics Program
contract #DAAL-03-90-C-0004.  
^*)