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.

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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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