The function $J(z)$ is defined by

$$J(z)=\int_{\mathcal{P}} t^{z-1}(1-t)^{b-1} d t$$

where $b$ is a constant (which is not an integer). The path of integration, $\mathcal{P}$, is the Pochhammer contour, defined as follows. It starts at a point $A$ on the axis between 0 and 1 , then it circles the points 1 and 0 in the negative sense, then it circles the points 1 and 0 in the positive sense, returning to $A$. At the start of the path, $\arg (t)=\arg (1-t)=0$ and the integrand is defined at other points on $\mathcal{P}$ by analytic continuation along $\mathcal{P}$.

(i) For what values of $z$ is $J(z)$ analytic? Give brief reasons for your answer.

(ii) Show that, in the case $\operatorname{Re} z>0$ and $\operatorname{Re} b>0$,

$$J(z)=-4 e^{-\pi i(z+b)} \sin (\pi z) \sin (\pi b) \mathrm{B}(z, b)$$

where $\mathrm{B}(z, b)$ is the Beta function.

(iii) Deduce that the only singularities of $\mathrm{B}(z, b)$ are simple poles. Explain carefully what happens if $z$ is a positive integer.