Inverse Fourier transform of a partial fraction decomposition looks to give a wrong answer

[Janko Slavič]

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?

[Kalevi Suominen]

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

[Janko Slavič]

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.