Gamma gamma gameleon

Let $\displaystyle \Gamma(s) \equiv \int_{0}^{\infty} e^{-t}t^{s-1} \ dt$ . Then, by integration by parts ,

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

$= |-e^{t}t^{s-1}|_{0}^{\infty} - \int_{0}^{\infty} -se^{-t}t^{s-1} \ dt$

$= 0 + s\int_{0}^{\infty} e^{-t}t^{s-1} \ dt$

$= s\Gamma(s)$

Now, $\Gamma(s) = (s-1)!$ finally seems to be satisfied at first glance, but this is not yet the case.

By induction , if $\Gamma(n) = (n-1)!$ for some base case $n$ , then equality holds for all $s \in \mathbb{Z}$ . (We don't need to prove the noninteger cases since we only care about $\Gamma$ corresponding to the original values of the factorial, ie positive integral.)

So, setting $n = 1$ ,

$\Gamma(1) = \int_{0}^{\infty} e^{-t} \ dt = |-e^{-t}|_{0}^{\infty} = 1$

as desired!

Thus, we require $\boxed{\text{A \& C}}$ .