Functions that interrelate the real part, $f^{\prime }$, and imaginary part, $f^{″}$, of the refractive index, $\hat{n}=n+\mathrm{i}k$, or relative permittivity, $\widehat{ϵ}={ϵ}^{\prime }+\mathrm{i}{ϵ}^{″}$, through $$f^{\prime }({\tilde{\nu }}_{\mathrm{a}})-f_{\mathrm{\infty }}^{\prime }=\frac{1}{\pi }P{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{f^{″}(\tilde{\nu })}{\tilde{\nu }-{\tilde{\nu }}_{\mathrm{a}}}\mathrm{d}\tilde{\nu }$$ $$f^{″}({\tilde{\nu }}_{\mathrm{a}})-f_{\mathrm{\infty }}^{\prime }=\frac{-1}{\pi }P{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{f^{\prime }({\tilde{\nu }}_{\mathrm{a}})-f_{\mathrm{\infty }}^{\prime }}{\tilde{\nu }-{\tilde{\nu }}_{\mathrm{a}}}\mathrm{d}\tilde{\nu }$$ where $P$ means that the principal part of the integral is taken at the singularity.