I hope this is the right place to report a bug in the Wolfram Alpha.
It looks like the Wolfram Alpha produces wrong answer for this question: "Fourier of 1/(t-i)" http://www.wolframalpha.com/input/?i=Fourier+of+1%2F%28t-i%29
It tells me that the Fourier transform of 1/(t-i) is zero for negative frequency, however the correct answer is exactly the opposite -- Fourier transform of 1/(t-i) is zero only for w > 0. (Because the Fourier transform is the integral of exp(-iwt)dt/(f-i) and for w > 0 it should be closed in the lower half of the complex plane which is free of poles in this case, the only pole is t = i).
Maybe the sign of the Fourier transform is defined oppositely in Alpha?
Wolfram Alpha is going to use Mathematica for the computation. The documented behavior for Mathematica is quite easy to find by web search. Here is the relevant link.
Click on Details and Options opener to see information on definition used and parameter settings.
Many thanks for the quick reply! Sorry for the stupid question. I see now, this is not a bug, the exponent in the Fourier transform is simply exp(+iwt), not exp(-iwt) as I assumed!