Re: 1/t System [hlp]
The Fourier Transform of 1/t can be easily obtained as follows (I'll use the notation in Mathematica):
F[1/t]=Integrate[Exp[-itw]/t,{t,-Inf,+Inf}]
=Integrate[Cos[tw]/t,{t,-Inf,+Inf}]-iIntegrate[Sin[tw]/t,{t,-Inf,+Inf}]
The first integral is zero when the Cauchy main value is applied. Actually, it's integral of an odd function. The second integral is a famous integral
iIntegrate[Sin[tw]/t,{t,-Inf,+Inf}]=Pi when w>0, and -Pi when w<0. Therefore,
F[1/t]=-i*Pi*[H(w)-H(-w)].