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We find the Bogoliubov coefficient from the tunneling boundary condition on a charged particle coupled to a static electric field $E_0 sech^2 (z/L)$ and, using the regularization scheme in Phys.
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ln Γ ( z ) = ∫ 0 ∞ d s s [ e − z s 1 − e − s − e − s 1 − e − s + ( z − 1 ) e − s ] \ln\Gamma(z)=\int_{0}^{\infty}\frac{ds}{s}\Bigl[\frac{e^{-zs}}{1-e^{-s}}-\frac{e^{-s}}{1-e^{-s}}+(z-1)e^{-s}\Bigr] ; see 8.341-3 of Ref. [ 24 ]
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