The gamma function is defined by
$$\Gamma(z)=\int_0^\infty t^{z-1}e^{-t}dt.$$ What made the mathematicians define it like this instead of $$\Gamma(z)=\int_0^\infty t^{z}e^{-t}dt.$$ This would make the gamma function a "real" generalization of the factorial function $\Gamma(n)=n!$
