Pochhammer contour
![](http://upload.wikimedia.org/wikipedia/commons/thumb/b/b6/Jordan-Pochhammer_curve.png/300px-Jordan-Pochhammer_curve.png)
In mathematics, the Pochhammer contour, introduced by Camille Jordan (1887) [1] and Leo Pochhammer (1890), 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.
Homologous to zero but not homotopic to zero
[edit]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.
![](http://upload.wikimedia.org/wikipedia/commons/thumb/c/c1/Pochhammer.homology2.svg/300px-Pochhammer.homology2.svg.png)
Applications
[edit]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.
Notes
[edit]- ^ Jordan (1887), pp. 243–244
References
[edit]- Jordan, C. (1887), Cours d'analyse, Tome III, Gauthier-Villars
- Pochhammer, L. (1890), "Zur Theorie der Euler'schen Integrale", Mathematische Annalen, 35 (4): 495–526, doi:10.1007/bf02122658
- Whittaker, E. T.; Watson, G. N. (1963), A Course of Modern Analysis, Cambridge University Press, ISBN 978-0-521-58807-2