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]
  • 2
    $\begingroup$ 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
  • 1
    $\begingroup$ Also as antiderivatives, Log[x] and Log[Abs[x]] differ by piecewise constants on the real line. $\endgroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.