On Friday, May 28, 2021 at 8:19:07 PM UTC+8 Emmanuel Charpentier wrote:
> This can be computed “by hand” using (one of) the textbook definition(s) :
>
> sage: var("omega, s")
> (omega, s)
> sage: integrate(sin(x^2)*e^(-I*s*x), x, -oo, oo)
> 1/2*sqrt(2)*sqrt(pi)*cos(1/4*s^2) - 1/2*sqrt(2)*sqrt(pi)*sin(1/4*s^2)
>
> Both sympy and giac have implementations of this transform :
>
> sage: from sympy import fourier_transform, sympify
> sage: fourier_transform(*map(sympify, (sin(x^2),x, s)))._sage_()
> 1/2*sqrt(2)*sqrt(pi)*(cos(pi^2*s^2) - sin(pi^2*s^2))
> sage: libgiac.fourier(*map(lambda u:u._giac_(), (sin(x^2), x, s))).sage()
> 1/2*sqrt(2)*sqrt(pi)*(cos(1/4*s^2) - sin(1/4*s^2))
>
> which do not follow the same definitions… But beware : they may be more or
> less wrong :
>
> sage: integrate(sin(x)*e^(-I*s*x), x, -oo, oo).factor()
> undef # Wrong
> sage: fourier_transform(*map(sympify, (sin(x),x, s)))._sage_()
> 0 # Wrong AND misleading
> sage: libgiac.fourier(*map(lambda u:u._giac_(), (sin(x), x, s))).sage()
> I*pi*dirac_delta(s + 1) - I*pi*dirac_delta(s - 1) # Better...
>
> BTW:
>
> sage: mathematica.FourierTransform(sin(x^2), x, s).sage().factor()
> 1/2*cos(1/4*s^2) - 1/2*sin(1/4*s^2)
> sage: mathematica.FourierTransform(sin(x), x, s).sage().factor()
> -1/2*I*sqrt(2)*sqrt(pi)*(dirac_delta(s + 1) - dirac_delta(s - 1))
>
> But what I got is different from yours:
sage: sage: var("omega,
s")
(omega, s)
sage: mathematica.FourierTransform(sin(x), x,
s).sage().factor()
-I*(dirac_delta(s + 1) - dirac_delta(s - 1))*Sqrt(1/2*pi)
BTW:
How to input the sage computation representation as the code style just
like what you've posted?
HY
> HTH,
>
> Le dimanche 23 mai 2021 à 03:22:06 UTC+2, hongy...@gmail.com a écrit :
>
>> It seems that all the Fourier transform methods implemented in sagemath
>> is numeric, instead of symbolic/analytic.
>>
>> I want to know whether there are some symbolic/analytic Fourier transform
>> functions, just as we can do in mathematica, in sagemath?
>>
>> I want to know if there are some symbolic/analytical Fourier transform
>> functions available in sagemath, just as the ones in mathematica?
>>
>> Regards,
>> HY
>>
>>
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