RE: Re: Generalized functions and integral transforms
- To: mathgroup at smc.vnet.net
- Subject: [mg34463] RE: [mg34445] Re: Generalized functions and integral transforms
- From: "DrBob" <majort at cox-internet.com>
- Date: Wed, 22 May 2002 02:46:38 -0400 (EDT)
- Reply-to: <drbob at bigfoot.com>
- Sender: owner-wri-mathgroup at wolfram.com
Here's an educational example:
InverseLaplaceTransform[E^(-2*k*y), y, x];
LaplaceTransform[%, x, y];
InverseLaplaceTransform[%, y, x];
LaplaceTransform[%, x, y];
InverseLaplaceTransform[%, y, x];
LaplaceTransform[%, x, y];
InverseLaplaceTransform[%, y, x];
LaplaceTransform[%, x, y];
{%%%%%%%%, %%%%%%%, %%%%%%, %%%%%, %%%%, %%%, %%, %} // InputForm //
TableForm
{DiracDelta[2*k - x]*UnitStep[-2*k + x], UnitStep[2*k]/E^(2*k*y),
DiracDelta[2*k - x]*UnitStep[2*k]*UnitStep[-2*k + x],
UnitStep[2*k]^2/E^(2*k*y),
DiracDelta[2*k - x]*UnitStep[2*k]^2*UnitStep[-2*k + x],
UnitStep[2*k]^3/E^(2*k*y),
DiracDelta[2*k - x]*UnitStep[2*k]^3*UnitStep[-2*k + x],
UnitStep[2*k]^4/E^(2*k*y)}
I've started with one of the terms summed in the third term in Name's
equality below. Each time I take the Inverse Laplace followed by the
Laplace, the term gains a power of UnitStep. I have no idea how this
affects the internals of things, but it can't be helpful.
In the following, Simplify in the third line and FullSimplify in the
fourth stops the madness. (Simplify in the fourth line isn't enough,
and simplifying earlier lines has no effect.)
InverseLaplaceTransform[E^(-2*k*y), y, x];
LaplaceTransform[%, x, y];
InverseLaplaceTransform[%, y, x] // Simplify;
LaplaceTransform[%, x, y] // FullSimplify;
{%%%%, %%%, %%, %} // InputForm // TableForm
{DiracDelta[2*k - x]*UnitStep[-2*k + x], UnitStep[2*k]/E^(2*k*y),
DiracDelta[2*k - x]*UnitStep[2*k, -2*k + x], UnitStep[k]/E^(2*k*y)}
By the way, TableForm gives strange output in both cases. (The output
was easier to read and pass to e-mail this way than any other way I
could find, though.)
Bobby Treat
-----Original Message-----
From: Name [mailto:mee at home.com.redline.ru]
To: mathgroup at smc.vnet.net
Subject: [mg34463] [mg34445] Re: Generalized functions and integral transforms
I think I finally figured it out -- this is just an error in
InverseLaplaceTransform:
Tanh[y] == (E^y-E^-y) / (E^y+E^-y) == 1+2Sum[(-1)^k E^(-2k y),
{k,1,Infinity}],
so InverseLaplaceTransform is
DiracDelta[x]+2Sum[(-1)^k DiracDelta[2k-x],{k,1,Infinity}],
while Mathematica gives
2Sum[(-1)^k DiracDelta[2k-x],{k,0,Infinity}].
Maxim
(correct e-mail address is m.r at prontomail.com; mailer acting up...)
DrBob wrote:
> My previous experience with the La Place transform functions is that
the
> examples in Help are the ONLY ones that work properly. (Good choice.)
>
> Bobby
>
> -----Original Message-----
> From: Name [mailto:mee at home.com.redline.ru]
To: mathgroup at smc.vnet.net
> Subject: [mg34463] [mg34445] Generalized functions and integral transforms
>
> Tanh[y]-
> LaplaceTransform[InverseLaplaceTransform[Tanh[y],y,x],x,y]//
> Simplify
>
> Out[1]=
> -1
>
> Where does it go wrong?
> Doing it step by step:
>
> InverseLaplaceTransform[Tanh[y], y, x]
>
> Out[2]=
> 2*Sum[(-1)^K$144*DiracDelta[2*K$144 - x], {K$144, 0, Infinity}]
>
> LaplaceTransform[(-1)^K$144*DiracDelta[2*K$144 - x], x, y]
>
> Out[3]=
> ((-1)^K$144*UnitStep[2*K$144])/E^(2*K$144*y)
>
> Why everything works as expected if we set UnitStep[0]==1/2?