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From what I understand the Gamma function was created to expand the factorial to the real number line (and complex plane). So why is it that $\Gamma(x)=(x-1)!$ and is not equal to $x!$, where transitioning from discrete to continuous is much simpler?

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marked as duplicate by Robert Wolfe, lhf, Parcly Taxel, Community Mar 1 '18 at 1:19

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  • $\begingroup$ In fact Gauss used $\Pi(n)=n!$ $\endgroup$ – MPW Mar 1 '18 at 0:57
  • $\begingroup$ Mostly historical. $\Gamma$ and $\Pi$ each have their advantages and disadvantages. They're related by a $1/z$ or a $z$, but why introduce more notation than you need? $\endgroup$ – Robert Wolfe Mar 1 '18 at 0:59