Functions based on the physical principle of causality that interconvert the real and imaginary parts of complex optical quantities when they are known over a sufficiently wide (strictly infinite) wavenumber range. They are frequently used to interconvert the real part, \(f'\), and imaginary part, \(f''\), of the refractive index, \(\boldsymbol{\hat{n}} = n + {\rm{i}}k\), the dielectric constant (relative permittivity), \(\boldsymbol{\hat{\epsilon}_{r}} = \epsilon_{r}' + \rm{i}\epsilon_{r}''\), or the logarithm of the complex reflection coefficient \(re^{i\varphi}\) through \[f'(\tilde{\nu}_{\rm{a}}) - f_{\infty}' = \frac{2}{\pi} P \int_{0}^{\infty} \frac{\tilde{\nu} f''(\tilde \nu)}{\tilde{\nu}^{2} - \tilde{\nu}_{\rm{a}}^{2}} {\rm{d}}\tilde{\nu}\] \[f''(\tilde{\nu}_{\rm{a}}) = \frac{-2\tilde{\nu}_{\rm{a}}}{\pi} P \int_{0}^{\infty} \frac{f'(\tilde{\nu}) - f_{\infty}'}{\tilde{\nu}^2 - \tilde{\nu}_{\rm{a}}^2} \rm{d}\tilde{\nu}\] where \(P\) means that the principal part of the integral is taken at the singularity.