The Beta function is defined for $\operatorname{Re} z>0$ by

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

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

Show that

$$\left(\frac{z+q}{z}\right) \mathrm{B}(z+1, q)=\mathrm{B}(z, q)$$

and explain how this result can be used to obtain the analytic continuation of $\mathrm{B}(z, q)$. Hence show that $\mathrm{B}(z, q)$ is analytic except for simple poles and find the residues at the poles.