What is Gamma Function ?

The basic definition of $\Gamma(x) = \int_{0}^1 (\ln{\frac{1}{t}})^{x-1}dt = \int_{0}^{\infty}t^{x-1}e^{-t}dt$

The 2nd form is easily obtained by substituting $\ln {\frac{1}{t}} = p$

Euler defined $\Gamma(x)$ another way

$\Gamma(x) = \lim_{r \to \infty}\Gamma_{r}(x) =\lim_{r \to \infty} \frac{r! r^x}{x(x+1)...(x+r)}$

One can use any of these forms to prove the formula

$\Gamma(x+1) = x\Gamma(x)$ and which eventually gives $\Gamma(x) = (x-1)!$

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