ok , my bad , here is the post again ..
i was trying to come with an easier way of finding the Hilbert transform of a function
, and here is what i did :
starting with Fourier transform of x :
now
where the contour encloses t .
"this is done by cauchy's integral formula"
=>
now
where
is the Hilbert transform of x .
=>
"z is replaced with s for conventional reasons"
now the program is to relate the last integral to the inverse laplace (or fourier !!) transform of
, by choosing the suitable contour(s) , and here is where i'm stuck !! so any help is appreciated .