# Inverse Fourier transformation discrepancy between Wolfram|Alpha's solution and mine [closed]

I searched for the inverse fourier transformation of
$$\mathcal{F(\omega)} = \frac{2}{(1+i\cdot \omega)^2 +4} \rightarrow i^2=-1$$

My solution (compliant with the solution from my textbook): $$\mathcal{F}^{-1}\{\mathcal{F}\{\omega\}\} = e^{-t}\cdot \sin{(2t)}\cdot \sigma {(t)}$$
Wolfram|Alpha: $$\mathcal{F}^{-1}\{\mathcal{F}\{\omega\}\} = i \sqrt{\frac{\pi }{2}} e^{(1-2 i) t} \left(-1+e^{4 i t}\right) \theta (-t)$$
Where does this difference come from and how can I make the output equal to my solution?

• If you want someone to explain you the mathematical equivalence and required assumptions, you should better ask it from Mathematics.SE.
– Johu
Commented Sep 17, 2018 at 15:14

Use FourierParameters-> {1, -1}:
InverseFourierTransform[2/((1 + I ω)^2 + 4), ω, t, FourierParameters -> {1, -1}] // FullSimplify

$$e^{-t} \theta (t) \sin (2 t)$$
More info about FourierParameters in the documentation.