We know asymptotic formula for gamma function as Stirling formula $$\Gamma(z+1) \approx F(z+1) = \sqrt{2\pi z}\left({\cfrac{z}{e}}\right)^z\cdot \left({1+\cfrac{1}{12z} +\cfrac{1}{288z^2}+...}\right)$$ The value of $\Gamma(0)$ and $\Gamma(-1)$ is of course infinity, but $$F(0) = -2\pi i\cdot (1+3.07...\cdot 10^{-3})$$ $$F(-1) = +2\pi i\cdot (1+3.9...\cdot 10^{-7})$$ is it a coincidence or something is connected to this?
2026-04-05 05:12:14.1775365934
Why Stirling formula at n = -1 and n= -2 is so close to 2*pi*i
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