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Machinato

Do not disturb my circles!

original contributions (listed by own intervention) :

  1. new elementary aproach to Gaussian integral $\int_{0}^{\infty}e^{-x^2}\mathrm{d}x$
  2. another proof of $\sum_{n=1}^{\infty}\frac1{n^3}\frac{\sinh\pi n\sqrt2-\sin\pi n\sqrt2}{{\cosh\pi n\sqrt2}-\cos\pi n\sqrt2}=\frac{\pi^3}{18\sqrt2}$, by the same argument $\sum_{n = 1}^{\infty}\{\coth (n\pi x) + x^{2}\coth(n\pi/x)\}/n^{3}$ is found in closed form
  3. the mean and the variance of the number of cycles in a random permutation of a given length n $\mu_n = H_n, \,\sigma_n^2 = H_n + H_n^{(2)}$ is derived

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