Solution
= Solution
\b[$J$ 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.