# [Maxima] laplace transform

Raymond Toy raymond.toy at stericsson.com
Thu Feb 26 11:13:29 CST 2009

```eric.reyssat at math.unicaen.fr wrote:
> Yes, the answer given by specint (all in lowercase) is true, thank you.
>
> But
>
> 1/ I still don't understand the error message given by laplace

Sounds like a bug in laplace
>
> 2/ when laplace is unable to compute the answer, it could (should ?) call
> specint

I think this has been discussed before.  But AFAIK, no one has actually
>
> 3/ Since I assumed s>0, I expected the answer to be given in a real form,
> I mean without %i in it. How can we "simplify" this answer to atan(2/s) ?
> Is there a quite general way to do it ?

I don't know how to get anything simpler than
atan2(4*s/(s^2+4),(s^2-4)/(s^2+4))/2.  It seems to me that we could at
least have atan2(4*s,s^2-4)/2, but I don't know how to get maxima to do
that.  I also don't know why you think the answer is atan(2/s).  Is that
the answer in some table of transforms?

Ray

>
> What I can do is only this :
> (%i56) u:specint(sin(2*t)/t*exp(-s*t),t);
> (%o56) %i*log((s-2*%i)/(s+2*%i))/2
> (%i57) trigrat(u);
> (%o57) atan2(4*s/(s^2+4),(s^2-4)/(s^2+4))/2
> which is obviously real for real s, but still quite complicated.
>
> Eric
>
>> Try specint, which handles Laplace transforms better:
>>
>> Specint(sin(2*t)/t*exp(-s*t),t) ->
>>
>> %i*log((s-2*%i)/(s+2*%i))/2
>>
>> (Don't know if that's right or not.)
>>
>> Ray
>>
>>
>> -----Original Message-----
>> From: maxima-bounces at math.utexas.edu
>> [mailto:maxima-bounces at math.utexas.edu] On Behalf Of
>> eric.reyssat at math.unicaen.fr
>> Sent: Thursday, February 26, 2009 9:15 AM
>> To: maxima at math.utexas.edu
>> Subject: [Maxima] laplace transform
>>
>> Hello,
>>
>> how comes that maxima pretends the following Laplace integral is
>> divergent ?
>> The function to integrate is not defined at 0, but the integral
>> converges for every positive s.
>> "integrate" doesn't find the answer.
>> The value of the integral should be atan(2/s), as checked for s=3 by
>> numerical computation with quad_qag :
>>
>> (%i1) build_info()\$
>> Maxima version: 5.17.0
>> Maxima build date: 19:8 12/4/2008
>> host type: i686-pc-mingw32
>> lisp-implementation-type: GNU Common Lisp (GCL)
>> lisp-implementation-version: GCL 2.6.8
>>
>> (%i2) display2d:false\$  assume(s>0)\$  laplace(sin(2*t)/t, t, s);
>> Integral is divergent
>>  -- an error.  To debug this try debugmode(true);
>> (%i5) integrate(sin(2*t)/t*exp(-s*t),t,0,inf);
>> (%o5) 'integrate(%e^-(s*t)*sin(2*t)/t,t,0,inf)
>> [atan(2/s)],numer;
>> (%o7) [0.58800060355057,2.2822993845756943E-10,285,0]
>> (%o8) [0.58800260354757]
>>
>>
>> Eric Reyssat
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>> Maxima at math.utexas.edu
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>>
>

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