Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 1 / 14B (Further Complex Methods)

$J$ is entire in $z$. Along the fixed Pochhammer contour, the continued logarithms of $t$ and $1-t$ are bounded, and the contour stays away from both branch points. The integrand $\exp ((z-1)\log t+(b-1)\log (1-t))$ is entire in $z$ with uniform bounds on each compact parameter set. Differentiation under the contour integral is therefore valid to every order.

Contract the loops in the Pochhammer contour onto the interval $(0,1)$. When $\mathrm{Re}z,\mathrm{Re}b>0$, the small-circle contributions tend to zero. The two clockwise circuits change the branches by $e^{-2\pi ib}$ and $e^{-2\pi iz}$, and the return circuits undo them. The four interval contributions sum toUsing the Beta function and $1-e^{-2\pi iu}=2i{e}^{-\pi iu}\sin (\pi u)$ gives

Because $b$ is not an integer, the coefficient $\sin (\pi b)$ is nonzero. The expression of $B$ in terms of the entire function $J$ can therefore have poles only at integral $z$, and at most simple ones. Positive integers are removable: $B(z,b)$ is already analytic there for $\mathrm{Re}b>0$, and analytic continuation in $b$ givesThus $J$ vanishes at those integers, cancelling the denominator zero. The Gamma function formula $B(z,b)=\Gamma (z)\Gamma (b)/\Gamma (z+b)$ identifies the remaining singularities:For nonintegral $b$, these residues are nonzero, so every listed singularity is a genuine simple pole.