Functions that interrelate the real part, \(f'\), and imaginary part, \(f''\), of the refractive index, \(\boldsymbol{\hat{n}} = n + {\rm{i}}k\), or relative permittivity, \(\boldsymbol{\hat{\epsilon}} = \epsilon' + \rm{i}\epsilon''\), through \[f'(\tilde{\nu}_{\rm{a}}) - f_{\infty}' = \frac{1}{\pi} P \int_{-\infty}^{\infty} \frac{f''(\tilde{\nu})}{\tilde{\nu} - \tilde{\nu}_{\rm{a}}}\rm{d}\tilde{\nu}\] \[f''(\tilde{\nu}_{\rm{a}}) - f_{\infty}' = \frac{-1}{\pi} P \int_{-\infty}^{\infty} \frac{f'(\tilde{\nu}_{\rm{a}}) - f_{\infty}'}{\tilde{\nu} - \tilde{\nu}_{\rm{a}}}\rm{d}\tilde{\nu}\] where \(P\) means that the principal part of the integral is taken at the singularity.