User:Turtlemanman

Physics Regents Grade

$\frac{{54{e^{13}}+94+2{e^{50}}\int_{e^{27}}^{e^{50}} \frac{1-lnx}{x^2}\,dx}}{2}$

The gamma function's connection to the factorial function.

$\Gamma(z) = \int_{0}^{\infty} t^{z-1}e^{-t}\,dt$ definition

$\Gamma(z) = [-t^{z-1}e^{-t}]_{0}^{\infty} - \int_{0}^{\infty} -e^{-t}(z-1)t^{z-2}\,dt$ integration by parts

$\Gamma(z) = 0 + (z-1)\int_{0}^{\infty} e^{-t}t^{(z-1)-1}\,dt$

Hence

$\Gamma(z+1) = z\int_{0}^{\infty} e^{-t}t^{z-1}\,dt = z\Gamma(z)$

It follows that, since $\textstyle \Gamma(1) = 1$ ,

$\textstyle \Gamma(z+1) = z\Gamma(z) = z(z-1)\Gamma(z-1) = z(z-1)(z-2)\Gamma(z-2) = ... = z!\Gamma(1) = z!$

Thus, in summary,

$\textstyle \Gamma(z+1) = z!$

Which shows the gamma function, $\textstyle \Gamma(z)$ , to be an extension of the factorial function, $\textstyle z!$ (for all natural numbers z).