Talk:Pochhammer contour

I do not believe the expression:

$$\left(1-e^{2\pi i \alpha}\right)\left(1-e^{2\pi i \beta}\right)B(\alpha,\beta)=\int_C t^{\alpha-1}(1-t)^{\beta-1}dt$$

is correct for all $$(\alpha,\beta)\in \mathbb{C}$$ and is only correct when $$(\alpha,\beta)\in \mathbb{Q}$$ and a particular set of determinations are traversed over the integration path. Therefore, I wish to discuss editing this article with a derivation of the more general expression

$$e^{2k\pi i\alpha} e^{2n\pi i\beta}\left(1-e^{2\pi i \alpha}\right)\left(1-e^{-2\pi i \beta}\right)B(\alpha,\beta)=\oint_{P(k,n)} t^{\alpha-1}(1-t)^{\beta-1}dt$$

which is valid for all $$(\alpha,\beta)\in \mathbb{C}$$ and clearly expresses the functional relationship between the left side and a particular choice of determinations on the right side with $$P(k,n)$$ defined accordingly.

However the first expression is widely published as the beta function continuation and I am suggesting all published references to the first expression are inadequate. I am by far, not an expert with this subject and are hoping others here more familiar with the pochhammer integral can offer their opinions about this proposal before I submit my derivation.

Jackmell (talk) 22:31, 29 November 2013 (UTC)

I believe what I should do in the interim period is to continue working on this problem and pursue consulting the primary references on this matter to see if I have made a mistake. 66.97.49.56 (talk) 19:41, 30 November 2013 (UTC)

I have come to realize this discrepancy I have about this problem is due in part to the fact that the pochhammer contour can take several forms. The particular description of the contour described above generates the cycles:

$$ f\to e^{-2\pi i b}f\to e^{-2\pi i b}e^{2\pi i a}f\to e^{-2\pi i b}e^{2\pi ib} e^{2\pi ia}f\to e^{-2\pi i a}e^{-2\pi i b}e^{2\pi ib} e^{2\pi i a}f,$$

or in a more compact form: $$(-1,+0,+1,-0)$$.

And that leads to the expression

$$\left(1-e^{2\pi i \alpha}\right)\left(1-e^{-2\pi i \beta}\right)B(\alpha,\beta)=\int_{P} t^{\alpha-1}(1-t)^{\beta-1}dt$$.

However, if we take the path $$(-1,-0,+1,+0)$$, we obtain:

$$\left(1-e^{2\pi i \alpha}\right)\left(1-e^{2\pi i \beta}\right)B(\alpha,\beta)=\int_{P} t^{\alpha-1}(1-t)^{\beta-1}dt$$.

Likewise, $$(+1,-0,-1,+0)$$ leads to the expression

$$\left(1-e^{-2\pi i \alpha}\right)\left(1-e^{2\pi i \beta}\right)B(\alpha,\beta)=\int_{P} t^{\alpha-1}(1-t)^{\beta-1}dt$$.

Those are all of course dependent upon starting the integration on the principal-valued sheet of the integrand. If the integration is started elsewhere, then we should in my opinion add the branch prefixes $$e^{2k\pi i\alpha} e^{2n\pi i\beta}$$.

Thus, the original expression

$$\left(1-e^{2\pi i a}\right)\left(1-e^{2\pi i b}\right)\beta(a,b)=\int_{P} t^{a-1}(1-t)^{b-1}dt,\quad (a,b)\in \mathbb{C},$$

is correct using the appropriate path to obtain positive exponents, and the more general form of the expression could be used otherwise.

— Preceding unsigned comment added by Jackmell (talk • contribs) 08:25, 29 January 2014 (UTC)

Hanging a picture
This curve gives us a way to hang a painting from two nails such that the removal of either one of the nails allows the painting to drop. In other words, the curve is hanging from the two of the nails but not from either of them individually. Anyone up for challenge to write this neatly and probably create a picture/animation of this? 130.234.20.37 (talk) 12:10, 28 February 2023 (UTC)