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\end{equation}  and analyse the integral  \begin{equation}  s_0=\int_{-\infty}^\infty \frac{1}{\sqrt[4]{\pi}}e^{-x^2/2}\mathrm{ln}\left(\frac{1}{\sqrt[4]{\pi}}e^{-x^2/2}\right)\;dx = \frac{\sqrt[4]{\pi}(2+\ln(\pi))}{2\sqrt{2}}  \end{equation}