TITLE: Why is the Gamma function off by 1 from the factorial?
QUESTION [8 upvotes]: Why didn't they define it as
$$ \tilde \Gamma(x) = \int_0^\infty t^x e^{-t} \, dt ?$$
Then the definition would have two less characters than the standard definition of $\Gamma(x)$, and we would have $\tilde \Gamma(n) = n!$ for $n$ a non-negative integer.  And this would save a lot of confusion.

REPLY [2 votes]: (Copied from https://mathoverflow.net/questions/20960/why-is-the-gamma-function-shifted-from-the-factorial-by-1)
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."
The book in question: 
https://books.google.com/books?id=5uLAoued_dIC&pg=PA7&source=gbs_toc_r&cad=4#v=onepage&q&f=false