Can we solve this strange functional equation? $ f(x+i\epsilon)-f(x-i\epsilon) = g(x) $
I believe that the solution is the Hilbert (finite part) transform of the function g(x) however I do not know it exactly.
I had thought taking in both sides the Fourier transform in tihs case i believe that $2 i F(p)\sin(p\epsilon)=G(p)$ so from this algebraic equation we could evaluate $f(x)$.