The expression

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


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]
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    $\begingroup$ I am not an expert on this, but looking at this question, it seems to me that the additive constant is irrelevant in any setting where the Fourier transform exists at all. $\endgroup$ – Lukas Lang Nov 12 '19 at 15:42
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    $\begingroup$ Also as antiderivatives, Log[x] and Log[Abs[x]] differ by piecewise constants on the real line. $\endgroup$ – Daniel Lichtblau Nov 12 '19 at 16:06

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.

  • $\begingroup$ It is essentially the same result as in the question, which is wrong $\endgroup$ – Anixx Nov 13 '19 at 7:06

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