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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

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    \(THIS\ FUNCTION\ "\<nakagami\>"\ IS\ A\ PROBABILITY\ DENSITY\ 
FUNCTION\ \
USED\ IN\ WIRELESS\ COMMUNICATION\ TO\ CALCULATE\ THE\ SIGNAL\ STRENGTH\ 
IN\ \
A\ FADING\ RADIO\ CHANNEL\), "\[IndentingNewLine]",
    \(nakagami[m_, \[CapitalOmega]_,
        z_] := \(\(\((m\/\[CapitalOmega])\)\^m\) \(z\^\(m - 1\)\) \
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    \(NOW\ I\ WANTED\ TO\ SQUARE\ THE\ LAPLACE\ TRANSFORM\ \((WHICH\ IS\ 
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EQUIVALENT\ TO\ SUNNING\ TOGETHER\ TWO\ RANDOM\ VARIABLES)\)\ AND\ I\ 
GET\
\[IndentingNewLine]\), "\[IndentingNewLine]",
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\((m\/\
\[CapitalOmega])\)\^m)\)\^2]\)}], "Input"],

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Cell[BoxData[
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Cell[BoxData[
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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]\), 
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\[Omega]\^a\
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Cell[BoxData[
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\((\(-a\) - \
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"]
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