# 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. Commented Nov 12, 2019 at 15:42
• Also as antiderivatives, Log[x] and Log[Abs[x]] differ by piecewise constants on the real line. Commented Nov 12, 2019 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 Commented Nov 13, 2019 at 7:06