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MathGroup Archive 1999

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Laplace vs. Inverse Laplace Transforms in Mathematica v4

  • To: mathgroup at smc.vnet.net
  • Subject: [mg19985] Laplace vs. Inverse Laplace Transforms in Mathematica v4
  • From: "Will Cooper" <wcooper1 at san.rr.com>
  • Date: Thu, 23 Sep 1999 23:26:29 -0400
  • Organization: Time Warner Cable of San Diego, CA
  • Sender: owner-wri-mathgroup at wolfram.com



Hello,
I can perform Laplace Transforms in Mathematica version 4 with no problems.

However, I'm having problems getting Mathematica to do certain inverse 
Laplace Transforms.

I attach a notebook showing both a successful and an unsucessful Inverse 
Laplace Transform calculation.

Please can you help.

Thanks,

Will Cooper.


                    Mathematica-Compatible Notebook

This notebook can be used on any computer system with Mathematica 4.0,
MathReader 4.0, or any compatible application. The data for the notebook 

starts with the line containing stars above.

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the following:

* Save the data starting with the line of stars above into a file
  with a name ending in .nb, then open the file inside the application;

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  clipboard, then use the Paste menu command inside the application.

Data for notebooks contains only printable 7-bit ASCII and can be
sent directly in email or through ftp in text mode.  Newlines can be
CR, LF or CRLF (Unix, Macintosh or MS-DOS style).

NOTE: If you modify the data for this notebook not in a Mathematica-
compatible application, you must delete the line below containing the
word CacheID, otherwise Mathematica-compatible applications may try to
use invalid cache data.

For more information on notebooks and Mathematica-compatible
applications, contact Wolfram Research:
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Notebook[{
Cell[BoxData[{
    \(THIS\ FUNCTION\ "\<nakagami\>"\ IS\ A\ PROBABILITY\ DENSITY\ 
FUNCTION\ \
USED\ IN\ WIRELESS\ COMMUNICATION\ TO\ CALCULATE\ THE\ SIGNAL\ STRENGTH\ 
IN\ \
A\ FADING\ RADIO\ CHANNEL\), "\[IndentingNewLine]",
    \(nakagami[m_, \[CapitalOmega]_,
        z_] := \(\(\((m\/\[CapitalOmega])\)\^m\) \(z\^\(m - 1\)\) \
\[ExponentialE]\^\(-\(\(m*z\)\/\[CapitalOmega]\)\)\)\/Gamma[m]\)}], 
"Input"],

Cell[BoxData[
    \(\(\(\[IndentingNewLine]\)\(THE\ LAPLACE\ TRANSFORM\ CAN\ BE\ \
SUCCESSFULLY\ OBTAINED\ AS\ SHOWN\ \(\(BELOW\)\(:\)\)\)\)\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
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Cell[BoxData[
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\((m\/\[CapitalOmega])\)\^m\)], "Output"]
}, Open  ]],

Cell[CellGroupData[{

Cell[BoxData[{
    \(NOW\ I\ WANTED\ TO\ SQUARE\ THE\ LAPLACE\ TRANSFORM\ \((WHICH\ IS\ 
\
EQUIVALENT\ TO\ SUNNING\ TOGETHER\ TWO\ RANDOM\ VARIABLES)\)\ AND\ I\ 
GET\
\[IndentingNewLine]\), "\[IndentingNewLine]",
    \(FullSimplify[\((\((\[Omega] + m\/\[CapitalOmega])\)\^\(-m\)\ 
\((m\/\
\[CapitalOmega])\)\^m)\)\^2]\)}], "Input"],

Cell[BoxData[
    \(\((\[Omega] + m\/\[CapitalOmega])\)\^\(\(-2\)\ m\)\ \((m\/\
\[CapitalOmega])\)\^\(2\ m\)\)], "Output"]
}, Open  ]],

Cell[BoxData[
    \(\(\(\[IndentingNewLine]\)\(NOW\ I\ WANTED\ TO\ FIND\ THE\ INVERSE\ 
\
LAPLACE\ TRNSFORM\)\)\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(InverseLaplaceTransform[%, \[Omega], z]\)], "Input"],

Cell[BoxData[
    \(\(\[ExponentialE]\^\(-\(\(m\ z\)\/\[CapitalOmega]\)\)\ z\^\(\(-1\) 
+ 2\ \
m\)\ \((m\/\[CapitalOmega])\)\^\(2\ m\)\)\/Gamma[2\ m]\)], "Output"]
}, Open  ]],

Cell[BoxData[
    \(\(SUCCESS!\)\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[{
    \(\[IndentingNewLine]\[IndentingNewLine]HOWEVER, \
    WHEN\ I\ TRY\ THIS\ ON\ A\ DIFFERENT\ PROBABILITY\ DISTRIBUTION\ 
FUNCTION\
\ "\<openloopnakrv\>", \
    I\ AM\ UNABLE\ TO\ GET\ THE\ INVERSE\ LAPLACE\ TRANSFORM\  - \ \
\[IndentingNewLine]\[IndentingNewLine]HOW\ CAN\ I\ GET\ THE\ INVERSE\ 
LAPLACE\
\ TRANSFORM\ OF\ THIS\ FUNCTION?\ 
\[IndentingNewLine]\[IndentingNewLine]ANY\ \
HELP\ WOULD\ BE\ GREATLY\ APPRECIATED . \
\[IndentingNewLine]\[IndentingNewLine]openloopnakrv[a_, m_, 
\[CapitalOmega]_,
              z_] := \(2\ a\^\(a + 1\/2\ \((\(-a\) + m)\)\)\ 
z\^\(\(-1\) + \
m\)\ \((m\/\[CapitalOmega])\)\^m\ \((\[CapitalOmega]\/\(m\ 
z\))\)\^\(1\/2\ \
\((\(-a\) + m)\)\)\ BesselK[\(-a\) + m, 2\ \ at a\ \ at \(\(m\ 
z\)\/\[CapitalOmega]\
\)]\)\/\(Gamma[a]\ Gamma[m]\)\[IndentingNewLine]\), 
"\[IndentingNewLine]",
    \(LaplaceTransform[openloopnakrv[a, m, \[CapitalOmega], z],
      z, \[Omega]]\)}], "Input"],

Cell[BoxData[
    \(\(1\/\(Gamma[a]\ Gamma[
              m]\)\) \((a\^\(a + 1\/2\ \((\(-a\) + m)\)\)\ 
\[Omega]\^\(\(-a\) \
- m\)\ \((m\/\[CapitalOmega])\)\^m\ \((\(a\ 
m\)\/\[CapitalOmega])\)\^\(1\/2\ \
\((\(-a\) - m)\)\)\ \((\[CapitalOmega]\/m)\)\^\(1\/2\ \((\(-a\) + 
m)\)\)\ \((\
\[Omega]\^m\ \((\(a\ m\)\/\[CapitalOmega])\)\^a\ Gamma[
                  a]\ Gamma[\(-a\) + m]\ Hypergeometric1F1[a,
                  1 + a -
                    m, \(a\ m\)\/\(\[Omega]\ \[CapitalOmega]\)] + 
\[Omega]\^a\
\ \((\(a\ m\)\/\[CapitalOmega])\)\^m\ Gamma[a - m]\ Gamma[
                  m]\ Hypergeometric1F1[m,
                  1 - a +
                    m, \(a\ m\)\/\(\[Omega]\ 
\[CapitalOmega]\)])\))\)\)], \
"Output"]
}, Open  ]],

Cell[BoxData[
    \(WORKS\ OK, \ \(\(NOW\)\(\ \)\(TRY\)\(\ \)\(THE\)\(\ 
\)\(INVERSE\)\(\ \)\
\(LAPLACE\)\(\ \)\(TRANSFORM\)\(\[IndentingNewLine]\)\)\)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(InverseLaplaceTransform[%, \[Omega], z]\)], "Input"],

Cell[BoxData[
    \(\(1\/\(Gamma[a]\ Gamma[
              m]\)\) \((a\^\(a + 1\/2\ \((\(-a\) + m)\)\)\ \((m\/\
\[CapitalOmega])\)\^m\ \((\(a\ m\)\/\[CapitalOmega])\)\^\(1\/2\ 
\((\(-a\) - \
m)\)\)\ \((\[CapitalOmega]\/m)\)\^\(1\/2\ \((\(-a\) + m)\)\)\ \
InverseLaplaceTransform[\[Omega]\^\(\(-a\) - m\)\ \((\[Omega]\^m\ 
\((\(a\ m\)\
\/\[CapitalOmega])\)\^a\ Gamma[a]\ Gamma[\(-a\) + m]\ 
Hypergeometric1F1[a,
                      1 + a -
                        m, \(a\ m\)\/\(\[Omega]\ \[CapitalOmega]\)] + \
\[Omega]\^a\ \((\(a\ m\)\/\[CapitalOmega])\)\^m\ Gamma[a - m]\ Gamma[
                      m]\ Hypergeometric1F1[m,
                      1 - a +
                        m, \(a\ m\)\/\(\[Omega]\ \[CapitalOmega]\)])\), 
\
\[Omega], z])\)\)], "Output"]
}, Open  ]],

Cell[BoxData[
    \(FAILS\ TO\ \(CALCULATE!\)\)], "Input"]
},
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