The Beta and Gamma functions are defined by

$$\begin{aligned} B(p, q) &=\int_{0}^{1} t^{p-1}(1-t)^{q-1} d t \\ \Gamma(p) &=\int_{0}^{\infty} e^{-t} t^{p-1} d t \end{aligned}$$

where $\operatorname{Re} p>0, \operatorname{Re} q>0$.

(a) By using a suitable substitution, or otherwise, prove that

$$B(z, z)=2^{1-2 z} B\left(z, \frac{1}{2}\right)$$

for $\operatorname{Re} z>0$. Extending $B$ by analytic continuation, for which values of $z \in \mathbb{C}$ does this result hold?

(b) Prove that

$$B(p, q)=\frac{\Gamma(p) \Gamma(q)}{\Gamma(p+q)}$$

for $\operatorname{Re} p>0, \operatorname{Re} q>0$