Gamma function

$$ \begin{aligned}&\Gamma(z)=\int_0^\infty\exp(-x)x^{z-1}\mathrm{d}x,\quad\mathrm{Re}(z)>0.\\\\&\Gamma(z+1)=z\Gamma(z).\end{aligned} $$

If $m$ a positive integer or zero then:

$$ \begin{aligned}&\Gamma(m+1)=m!,\quad\frac{1}{\Gamma(-m)}\to0. \\&\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}. \\&\Gamma(z)\Gamma(1-z)=\frac{\pi}{\sin(\pi z)}. \\&\Gamma(2z )=\frac{2^{2z-1}}{\sqrt{\pi}}\Gamma(z)\Gamma\left(z+\frac12\right). \end{aligned} $$

The beta function $B(\alpha,\beta)$ is defined by:

$$ \begin{aligned}B(\alpha,\beta)&=\frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)}.\end{aligned} $$

$$ B(\alpha,\beta)=\int_0^1x^{\alpha-1}(1-x)^{\beta-1}\mathrm{d}x,\quad\mathrm{Re}(\alpha)>0,\mathrm{Re}(\beta)>0. $$