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Why are the following two formulas for the phase of the reflection coefficient equivalent (according e.g. to [1])?

\begin{align} \Theta(\omega_0) & = - \frac{\omega_0}{\pi}P\int_{0}^{\infty}\frac{\ln R(\omega)}{\omega^2 - \omega_0^2} d\omega \\ \Theta(\omega_0) & = - \frac{\omega_0}{\pi}\int_{0}^{\infty}\frac{\ln( R(\omega)/R(\omega_0))}{\omega^2 - \omega_0^2} d\omega \end{align}

My feeling is that the proof probably involves subtracting out the singularity, but I haven't been able to reach the conclusion that the formulas are equivalent using that approach.

  1. Nilsson, P. O.; Munkby, L.; Investigation of errors in the Kramers-Kronig analysis of reflectance data. Phys. Kondens. Mater. 10, 290 (1969).
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