In mathematics, the Pochhammer contour, introduced by Jordan (1887), pp. 243–244 and , is a contour in the

complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by th ...

with two points removed, used for

contour integration In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane. Contour integration is closely related to the calculus of residues, a method of complex analysis. ...

. If ''A'' and ''B'' are loops around the two points, both starting at some fixed point ''P'', then the Pochhammer contour is the

commutator In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, a ...

''ABA''⁻¹''B''⁻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

when it is considered in the

fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...

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

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

winding number In mathematics, the winding number or winding index of a closed curve in the plane around a given point is an integer representing the total number of times that curve travels counterclockwise around the point, i.e., the curve's number of tu ...

about any point is 0 despite the fact that within the doubly punctured plane it cannot be shrunk to a single point. Pochhammer

Applications

The

beta function In mathematics, the beta function, also called the Euler integral of the first kind, is a special function that is closely related to the gamma function and to binomial coefficients. It is defined by the integral : \Beta(z_1,z_2) = \int_0^1 t^(1 ...

is given by Euler's integral :

\displaystyle \Beta(\alpha,\beta)=\int_0^1 t^(1-t)^ \, dt

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-e^)(1-e^)\Beta(\alpha,\beta) =\int_C t^(1-t)^ \, dt.

The contour integral converges for all values of ''α'' and ''β'' and so gives the

analytic continuation In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a ne ...

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

Notes

References

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