Computing some Principal Value integrals without Residues and Applications on Hilbert Transform and Fourier Transform

This article proposes a new approach in the treatment of the Hilbert transform and some cases of the Fourier transform whose improper integrals are principal values. This approach may be useful for teaching these issues to undergraduate engineering students. Traditional literature of Complex Analysis deals with these transformation integrals with the Cauchy-Goursat theorem and the residues calculation technique. In this new approach, instead of residues, we use an intuitive result about complex line integrals of continuous complex functions that resembles the delta of Dirac.