I'm trying to calculate the real part of a response function, \chi = \chi' + j \chi'' , from its imaginary part using the following relationships

\chi'(\omega) =\frac{1}{\pi} P.V. \int^{\infty}_{-\infty} \frac{\omega' \chi''(\omega')}{\omega'^2 - \omega^2} d\omega', \quad \chi''(\omega) =-\frac{\omega}{\pi} P.V. \int^{\infty}_{-\infty} \frac{\chi'(\omega')}{\omega'^2 - \omega^2} d\omega'

\chi''(\omega) = \frac{A \gamma}{\omega}\frac{1}{ (\omega - \omega_0)^2 + \gamma^2}

I have attached my derivation in the pdf file. However the forward and reverse derivations do not yield the same result. Can someone kindly point me out whether I'm applying the Cauchy residue theorem incorrectly.