I am trying to replicate this:
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
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))
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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