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, $f^{\prime }$, and imaginary part, $f^{″}$, of the refractive index, $\hat{n}=n+\mathrm{i}k$, the dielectric constant (relative permittivity), ${\widehat{ϵ}}_r={ϵ}_r^{\prime }+\mathrm{i}{ϵ}_{\mathrm{r}}^{″}$, or the logarithm of the complex reflection coefficient $r{e}^{i\phi }$ through $$f^{\prime }({\tilde{\nu }}_{\mathrm{a}})-f_{\mathrm{\infty }}^{\prime }=\frac{2}{\pi }P{\int }_0^{\mathrm{\infty }}\frac{\tilde{\nu }{f}^{″}(\tilde{\nu })}{{\tilde{\nu }}^2-{\tilde{\nu }}_{\mathrm{a}}^2}\mathrm{d}\tilde{\nu }$$ $$f^{″}({\tilde{\nu }}_{\mathrm{a}})=\frac{-2{\tilde{\nu }}_{\mathrm{a}}}{\pi }P{\int }_0^{\mathrm{\infty }}\frac{f^{\prime }(\tilde{\nu })-f_{\mathrm{\infty }}^{\prime }}{{\tilde{\nu }}^2-{\tilde{\nu }}_{\mathrm{a}}^2}\mathrm{d}\tilde{\nu }$$ where $P$ means that the principal part of the integral is taken at the singularity.