# Fourier transform gives wrong result

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]

• 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. – Lukas Lang Nov 12 '19 at 15:42
• Also as antiderivatives, Log[x] and Log[Abs[x]] differ by piecewise constants on the real line. – 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.

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