Residue Evaluation of g

     (0.1)

This is also

       (0.2)

The back transform is

             (0.3)

For x > 0 the integral needs to be taken in the negative imaginary f plane.  This contains a pole at  
with residue
  
and a second pole at

with residue
  
So that 2πj times the sum of the residues is

           (0.4)
The symmetry of (0.3) gives the same result with x à |x| for x < 0.