Re: LaplaceTransform[SquareWave[]] ??
- To: mathgroup at smc.vnet.net
- Subject: [mg107324] Re: LaplaceTransform[SquareWave[]] ??
- From: Geico Caveman <spammers-go-here at spam.invalid>
- Date: Tue, 9 Feb 2010 02:45:30 -0500 (EST)
- References: <hkoich$t3c$1@smc.vnet.net> <hkp1n0$7fb$1@smc.vnet.net>
On 2010-02-08 05:57:04 -0700, "Nasser M. Abbasi" <nma at 12000.org> said:
> On Feb 8, 12:35 am, Geico Caveman <spammers-go-h... at spam.invalid>
> wrote:
>> In[1]:=LaplaceTransform[SquareWave[t], t, s]
>> Out[1]:=LaplaceTransform[SquareWave[t], t, s]
>>
>> That is a pretty standard Laplace transform. What gives ?
>>
>> LaplaceTransform[l'[t]] is fine.
>>
>> I am using Mathematica 7 on Mac OSX Snow Leopard.
>
>
> My guess is that you should not use SquareWave as is for this sort of
> thing. There seems to be some limit:
>
> Integrate[SquareWave[t],{t,0,1}]
> Integrate[SquareWave[t],{t,0,2}]
> ....
> Integrate[SquareWave[t],{t,0,99}]
>
> All of the above gives 0 as expected. But
>
> Integrate[SquareWave[t],{t,0,100}]
>
> failes with the error
>
> "Integrate::mpwc: {At Line = 22, the input
> was:,Integrate[SquareWave[t],{t,0,100}],Integrate} was unable to
> convert {At Line = 22, the input was:,Integrate[SquareWave[t],{t,
> 0,100}],Floor[t]} to Piecewise, because the required number {At Line
> 22, the input was:,Integrate[SquareWave[t],{t,0,100}],101} of
> piecewise cases sought exceeds the internal limit $MaxPiecewiseCases
> {At Line = 22, the input was:,Integrate[SquareWave[t],{t,0,100}],100}.
>>> "
>
> So, there is an internal limit $MaxPiecewiseCases
>
> In[24]:= $MaxPiecewiseCases
> Out[24]= 100
>
> So, my guess is that the Laplace integral
>
> Integrate[SquareWave[t]*Exp[-s t],{t,0,Infinity}]
>
> Was failing internally on this limit. Because when I do
>
> Integrate[SquareWave[t]*Exp[-s t],{t,0,99}]
> It works, but
> Integrate[SquareWave[t]*Exp[-s t],{t,0,100}]
> it fails
> and infinity is more than 100, so LaplaceTransform[] must have hit
> this internally somewhere, but did not report it?
>
> But the square wave is a simply function, it flips between 1 and -1
> with period 1, so we can directly do the Laplace integral on it.
>
> Split the integral into 2 parts. One part does the part when the wave
> is +1, and the second integral does the part when the wave is -1. Then
> add the 2 integrals together, and take the limit to infinity, I get
>
> Tanh[s/4]/s
>
> Here is the code:
>
> Clear[int1,int2,t,k,M,s]
> int1=Integrate[Exp[-s t],{t,k,k+1/2}];
> int2=Integrate[Exp[-s t],{t,k+1/2,k+1}];
> Sum[ int1-int2,{k,0,M}]//FullSimplify;
> Assuming[Element[s,Reals]&&s>0,Limit[%,M->Infinity]]
>
> Out[112]= Tanh[s/4]/s
>
> Is this is correct Laplace transform for square wave?
>
> --Nasser
Thanks for the detailed response.
I was planning on using the SquareWave as the forcing function for a
Riccati equation and then transform to s-domain to see if I could
simplify something (like drop a term or so) to get an approximate
solution.
I know what the transform for the -1,1 and 0,1 square wave is :)