The Beta function is defined for $\operatorname{Re}(z)>0$ as

$$B(z, q)=\int_{0}^{1} t^{q-1}(1-t)^{z-1} d t, \quad(\operatorname{Re}(q)>0)$$

and by analytic continuation elsewhere in the complex $z$-plane.

Show that:

(i) $(z+q) B(z+1, q)=z B(z, q)$;

(ii) $\Gamma(z)^{2}=B(z, z) \Gamma(2 z)$.

By considering $\Gamma\left(z / 2^{m}\right)$ for all positive integers $m$, deduce that $\Gamma(z) \neq 0$ for all $z$ with $\operatorname{Re}(z)>0$.