## Dirichlet’s Theorem and Liouville’s Extension of Dirichlet’s Theorem

Topic Beta & Gamma functions Statement $\int \int \int_{V} x^{l-1} y^{m-1} z^{n-1} dx dy ,dz = \frac { \Gamma {(l)} \Gamma {(m)} \Gamma {(n)} }{ \Gamma{(l+m+n+1)} }$ , where V is the region given by $x \ge 0 y \ge 0 z \ge 0 x+y+z \le 1$ . [wc_divider style="solid" line="single" margin_top="" margin_bottom=""] Brief Theory on Gamma and Beta Functions Gamma Function If we consider the integral $I =\displaystyle{\int_0^{\infty}} e^{-t} t^{a-1} \mathrm dt$ , it…