the normalization is not a problem. Dear Suominen, than you for the try,
but this was not the point.
Sympy uses the unitary, ordinary-frequency inverse Fourier transform and
there should be not 1/2pi.
The problem is much deeper; that final result (without the normalization)
is different from the theoretical result when the assumption Re[lambda]>0
or Re[lambda]<0 and t>0 or t<0 are used.
Hope for help.
On Tuesday, April 10, 2018 at 1:44:29 PM UTC+2, Kalevi Suominen wrote:
>
> There are two common definitions of inverse Fourier transform depending on
> how the frequency variable is interpreted. (See
> https://en.wikipedia.org/wiki/Fourier_transform#Other_conventions) The
> stackexchange post is using the angular frequency ω that differs by the
> factor 2π from frequency variable assumed by SymPy. The transforms are also
> differently normalized. That may explain the differing results.
>
> Kalevi Suominen
>
> On Tuesday, April 10, 2018 at 9:16:57 AM UTC+3, Janko Slavič wrote:
>>
>> I am trying to replicate this:
>>
>> https://math.stackexchange.com/questions/2280196/inverse-fourier-transform-of-a-partial-fraction-decomposition
>> with sympy.
>>
>> With Mathematica I get just the same behavior as the above theory
>> suggest.
>>
>> With sympy I it looks wrong. At t<0, I get the result close to the
>> correct one for t>0. (I was not able to include all the assumptions in
>> sympy).
>>
>> Here is the sympy code:
>>
>> import sympy as sym
>> sym.init_printing()
>> ω = sym.symbols('omega', real=True, positive=True)
>> R, λ = sym.symbols('R, lambda', complex=True)
>> t = sym.symbols('t', real=True, positive=True)
>> α = R/(sym.I*ω-λ)+sym.conjugate(R)/(sym.I*ω-sym.conjugate(λ))
>> α
>> sym.inverse_fourier_transform(α, ω, -t)
>>
>>
>> and the Mathematica:
>>
>> a = InverseFourierTransform[ R/(I omega - lambda) + Conjugate[R]/(I
>> omega - Conjugate[lambda]), omega, t,
>> FourierParameters -> {1, -1}]
>> Simplify[a, {Re[lambda] < 0, t > 0}]
>>
>> Is the sympy result really wrong?
>>
>
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