= Thermal ratio of Hawking Bogoliubov coefficients
{title2=$|\beta|^2/|\alpha|^2=e^{-2\pi\omega/\kappa}$}
For the <Hawking exponential ray map>, regulated integrals $\int_0^\infty x^{ia}e^{-(\epsilon\pm i\omega')x}\,dx=\Gamma(1+ia)(\epsilon\pm i\omega')^{-1-ia}$ have squared-modulus ratio $e^{-2\pi a}$ as $\epsilon\downarrow0$, 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.
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