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?