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I would like to do a Hilbert transformation in Mathematica on a function. However, it does not seem to give a right result.

The Hilbert transform is given by

enter image description here

So I did :

 Integrate[Sin[t]/(k - t), {t, -Infinity, Infinity}];

but it does not work

The only reference I found online is this :

https://www.wolframalpha.com/input?i=Hilbert+transform+sinx

Does anyone have an idea on how to do this transform as a Mathematica code?

Thanks

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