Pochhammer contour

If A and B are loops around the two points, both starting at some fixed point P, then the Pochhammer contour is the commutator ABA−1B−1, where the superscript −1 denotes a path taken in the opposite direction.

With the two points taken as 0 and 1, the fixed basepoint P being on the real axis between them, an example is the path that starts at P, encircles the point 1 in the counter-clockwise direction and returns to P, then encircles 0 counter-clockwise and returns to P, after that circling 1 and then 0 clockwise, before coming back to P. The class of the contour is an actual commutator when it is considered in the fundamental group with basepoint P of the complement in the complex plane (or Riemann sphere) of the two points looped.

Within the doubly punctured plane this curve is homologous to zero but not homotopic to zero.

The beta function is given by Euler's integral provided that the real parts of α and β are positive, which may be converted into an integral over the Pochhammer contour C as The contour integral converges for all values of α and β and so gives the analytic continuation of the beta function.

A similar method can be applied to Euler's integral for the hypergeometric function to give its analytic continuation.