Fourier transform gives wrong result

Question

The expression

I InverseFourierTransform[FourierTransform[1/t, t, w]/w, w, x]//FullSimplify

gives

EulerGamma + Log[Abs[x]]

while the correct result should be

EulerGamma - I Pi + Log[x]

the same as of the following:

f[n_, s_] := ((-1)^n n!)/s^(n + 1)
Limit[1/2 f[-1 + h, s] + 1/2 f[-1 - h, s], h -> 0]

Answer 1

By using the additional Assumptions -> in the script leads to

I*InverseFourierTransform[FourierTransform[1/t, t, w]/w, w, x, Assumptions -> x>=0]//FullSimplify

giving the result

EulerGamma + Log[x] + I*Pi*(1 - Sign[x])/2

The last portion of the result can, essentially, be ignored.