The Beta function is defined by

$$B(p, q)=\int_{0}^{1} t^{p-1}(1-t)^{q-1} d t$$

for $\operatorname{Re} p>0$ and $\operatorname{Re} q>0$.

(a) Prove that $B(p, q)=B(q, p)$ and find $B(1, q)$.

(b) Show that $(p+z) B(p, z+1)=z B(p, z)$.

(c) For each fixed $p$ with $\operatorname{Re} p>0$, use part (b) to obtain the analytic continuation of $B(p, z)$ as an analytic function of $z \in \mathbb{C}$, with the exception of the points $z=$ $0,-1,-2,-3, \ldots$

(d) Use part (c) to determine the type of singularity that the function $B(p, z)$ has at $z=0,-1,-2,-3, \ldots$, for fixed $p$ with $\operatorname{Re} p>0$.