Why is gamma function defined such that $\Gamma (n)=(n-1)!$ rather then $\Gamma(n)=n!$, The latter ssems far more logical.
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1$\begingroup$ Can this, this or math.stackexchange.com/questions/2244134/… help? $\endgroup$– Sewer KeeperJun 5, 2020 at 11:45
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An excerpt from this link:
From Riemann's Zeta Function, by H. M. Edwards, available as a Dover paperback, footnote on page 8: "Unfortunately, Legendre subsequently introduced the notation $\Gamma(s)$ for $\Pi(s−1)$.Legendre's reasons for considering (n−1)! instead of n! are obscure (perhaps he felt it was more natural to have the first pole at $s=0$ rather than at $s=−1$) but, whatever the reason, this notation prevailed in France and, by the end of the nineteenth century, in the rest of the world as well. Gauss's original notation appears to me to be much more natural and Riemann's use of it gives me a welcome opportunity to reintroduce it."
