I'ld like to calculate the PV of an integral with the form$$ \tilde{G}_l(\omega) = -\frac{2\omega}{\pi} PV\int_0^\infty \frac{\tilde{G}_d(\omega^\prime)}{\omega^2 - {\omega^\prime}^2}d\omega^\prime$$in MATLAB.  It is not obvious to me how to do this.  I've tried  using factorization to change the power in the denominator to allow for use with the Hilbert function in MATLAB, but that hasn't worked.  Any insight would be appreciated.$\tilde{G}_d(\omega)$ is an even function so I feel like I could do a contour integration and pick up the residue in the right-half plane, but I would rather do it numerically, as $\tilde{G}_d(\omega)$ exists more completely as a numerical function than an analytic one.
Numerical Principal Value Integration - Hilbert like
quadrature