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 .