The Beta function, denoted by $B\left(z_{1}, z_{2}\right)$, is defined by

$$B\left(z_{1}, z_{2}\right)=\frac{\Gamma\left(z_{1}\right) \Gamma\left(z_{2}\right)}{\Gamma\left(z_{1}+z_{2}\right)}, \quad z_{1}, z_{2} \in \mathbb{C}$$

where $\Gamma(z)$ denotes the Gamma function. It can be shown that

$$B\left(z_{1}, z_{2}\right)=\int_{0}^{\infty} \frac{v^{z_{2}-1} d v}{(1+v)^{z_{1}+z_{2}}}, \quad \operatorname{Re} z_{1}>0, \operatorname{Re} z_{2}>0$$

By computing this integral for the particular case of $z_{1}+z_{2}=1$, and by employing analytic continuation, deduce that $\Gamma(z)$ satisfies the functional equation

$$\Gamma(z) \Gamma(1-z)=\frac{\pi}{\sin \pi z}, \quad z \in \mathbb{C} .$$