Unexpected Fourier transform (including time shifting property) result in Mathematica/WolframAlpha?

Questions
,
1

This is what Mathematica produces for the Fourier transform of Sin[w0 t]:

Capture01.JPG

Ignoring the Sqrt(Pi/2) factor, the Fourier transform of Sin[w0 t] is:

Capture02.JPG

(source: Fourier Transform of the Sine and Cosine Functions & many others).
Notice the sign difference of the i (or j, in the engineering world) in the above expressions.
Also, quite disturbingly, the time shifting property of the Fourier transform appears to be wrong in WolframAlpha, but nevertheless consistent with the Mathematica behavior reported above:

Capture03.JPG
Capture03.JPG284×300 10.8 KB

compare with:

Capture04.JPG

What am I doing wrong? I sure hope this is not a bug.

2

Check the documentation for the FourierParameters option defaults and alternative settings.

3

Thank you - I can breathe easier now, although I’m still a little shaken, was not aware of the FourierParameters option and different definitions:

Capture05.JPG
Capture05.JPG976×605 57.9 KB

Much appreciated, we can close this.