In mathematics, the Pochhammer contour, introduced by [1] and, is a contour in the complex plane with two points removed, used for contour integration. 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. When it comes to taking contour integrals, moving basepoint from P to another choice Q makes no difference to the result, since there will be cancellation of integrals from P to Q and back.
Within the doubly punctured plane this curve is homologous to zero but not homotopic to zero. Its winding number about any point is 0 despite the fact that within the doubly punctured plane it cannot be shrunk to a single point.
The beta function is given by Euler's integral
\displaystyle
| 1 | |
| \Beta(\alpha,\beta)=\int | |
| 0 |
t\alpha-1(1-t)\beta-1dt
provided that the real parts of α and β are positive, which may be converted into an integral over the Pochhammer contour C as
\displaystyle(1-e2\pi)(1-e2\pi)\Beta(\alpha,\beta)=\intCt\alpha-1(1-t)\beta-1dt.
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.