Pochhammer contour
= Pochhammer contour
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A commutator contour around two branch points cancels the total <monodromy> of its integrand. For the contour circling $1,0$ clockwise and then $1,0$ anticlockwise, with initial positive-real branches, the <Beta function> obeys
$$
\int_Pt^{z-1}(1-t)^{b-1}\,dt=(1-e^{-2\pi iz})(1-e^{-2\pi ib})B(z,b).
$$
The contour integral is <entire> in $z$, giving a <meromorphic continuation> of the <Beta function>.