Thermal ratio of Hawking Bogoliubov coefficients ($|\beta {|}^2/|\alpha {|}^2=e^{-2\pi \omega /\kappa }$)

For the Hawking exponential ray map, regulated integrals ${\int }_0^{\infty }{x}^{ia}{e}^{-(ϵ\pm i{\omega }^{\prime })x}dx=\Gamma (1+ia)(ϵ\pm i{\omega }^{\prime }{)}^{-1-ia}$ have squared-modulus ratio $e^{-2\pi a}$ as $ϵ\downarrow 0$, where $a=\omega /\kappa$. Combining this with the canonical identities for a bosonic Bogoliubov transformation gives the bosonic occupation $[e^{2\pi \omega /\kappa }-1{]}^{-1}$. Continuum modes require wave-packet normalization to interpret finite particle counts.