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Inverse/LaplaceTransform inconsistency?
*To*: mathgroup at smc.vnet.net
*Subject*: [mg35180] Inverse/LaplaceTransform inconsistency?
*From*: michael_chang86 at hotmail.com (Michael Chang)
*Date*: Fri, 28 Jun 2002 02:31:36 -0400 (EDT)
*Sender*: owner-wri-mathgroup at wolfram.com
Hi everyone!
While running Mathematica 4.1 on WindozeXP, I've run across what
appears to be an inconsistency when using LaplaceTransform and
InverseLaplaceTransform on a very simple example.
I defined two square waves f[x] and g[x] as
In[1]:= f[x_]=mf(UnitStep[x]-UnitStep[x-a]);
In[2]:= g[x_]=mg(UnitStep[x-(c-b/2)]-UnitStep[x-(c+b/2)]);
The respective square waves are supposed to be centered about a/2 and
c, with widths of a and b, and magnitudes of mf and mg. Since I'll be
using LaplaceTransform, for simplicity, one can assume that
a,b,c,mf,mg are all >0.
Anyways, since I want to convolve these two (simple) functions, I've
decided to calculate the expressions
In[3]:= tff=LaplaceTransform[f[x],x,s]
In[4]:= tfg=LaplaceTransform[g[x],x,s]
which indeed yield correct results. In addition, the expression
returned by
In[5]:= soln1[x_]=InverseLaplaceTransform[tff*tfg,s,x]
is correct. (For a simple check, I tried
{a->1,b->2,c->10,mf->1,mg->1}, and verified graphically that my plot
was indeed a trapezoid.) However,
In[6]:= soln2[x_]=InverseLaplaceTransform[Apart[tff*tfg],s,x]
returns a completely different (and WRONG (IMHO)) answer involving
funny things such as UnitStep[]^2. This occurs even though, as
expected,
In[7]:= Simplify[Apart[tff*tfg]==tff*tfg]
Out[7]: True
Why is this? Am I in error here? Or is Mathematica in error?
Many thanks in advance!
Michael
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