Timeline for Why is the Gamma function shifted from the factorial by 1?
Current License: CC BY-SA 2.5
29 events
| when toggle format | what | by | license | comment | |
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| May 3 at 14:34 | comment | added | Terry Tao | As per the Hadamard factorization theorem, the Gamma/Pi function is largely determined by its poles (the nonpositive integers/negative integers). So it’s related to the perennial question of whether one prefers the natural numbers to start at 0 or 1. | |
| Oct 12, 2019 at 18:45 | comment | added | Hyperplane |
Why are we using $\pi$ instead of $\tau$? ¯\_(ツ)_/¯
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| Jun 10, 2017 at 17:43 | answer | added | Michael Hardy | timeline score: 5 | |
| Jun 10, 2017 at 17:13 | comment | added | Michael Hardy | I think Pietro Majer's answer is the best one so far. | |
| Sep 16, 2014 at 4:43 | answer | added | Lucian | timeline score: 3 | |
| Apr 26, 2014 at 18:12 | answer | added | GuyR | timeline score: 3 | |
| Mar 20, 2011 at 16:33 | answer | added | GH from MO | timeline score: 65 | |
| Mar 19, 2011 at 23:26 | answer | added | DavidLHarden | timeline score: 14 | |
| Oct 18, 2010 at 12:00 | answer | added | Greg Kuperberg | timeline score: 55 | |
| Oct 18, 2010 at 9:46 | answer | added | Anixx | timeline score: 32 | |
| May 26, 2010 at 19:18 | answer | added | Michael Hardy | timeline score: 10 | |
| May 26, 2010 at 7:23 | answer | added | Pietro Majer | timeline score: 25 | |
| Apr 11, 2010 at 8:25 | comment | added | Kevin Buzzard | For what it's worth, I always felt that the answer to this question would be embedded somehow in Tate's thesis and the theory of motives and general gamma factors at infinity, but having tried to get on top of this stuff the only thing I convinced myself of was that there is no "one correct gamma function"---one uses $\Gamma(s/2)$ and $\Gamma((s+1)/2)$ and $\Gamma(s)$ and it's not clear to me why any of these are more fundamental than any other. | |
| Apr 11, 2010 at 3:25 | comment | added | S. Carnahan♦ | I would argue that both functional equations you listed are really arguments in favor of $\Gamma$ over $\Pi$, especially since the symmetric form of the functional equation for $\zeta$ (using the $\xi$ function) is even more compact. Emerton's Haar measure argument is quite strong when viewed in the broader context of Tate's thesis, where you need to consider homogeneous measures at each place. Other points in favor of $\Gamma$ include the beta function definition, and the Mellin transform formula for $\zeta$ in terms of $\theta$. | |
| Apr 11, 2010 at 2:22 | answer | added | castal | timeline score: 4 | |
| Apr 10, 2010 at 23:28 | vote | accept | Kevin Casto | ||
| Apr 10, 2010 at 23:23 | comment | added | Kevin Casto | Emerton, that's exactly the explanation my number theory professor just gave me yesterday. Still, I don't think it's very satisfying given everything else. | |
| Apr 10, 2010 at 23:15 | comment | added | Emerton | Note: the integral for the $\Gamma$ function is not so unreasonable, if you think of it as an integral of $t^z e^{-t}\frac{dt}{t}$, i.e. an integral transform over $\mathbb R^{\times}_{>0}$ with respect to multiplicative Haar measure. | |
| Apr 10, 2010 at 22:55 | history | edited | Kevin Casto | CC BY-SA 2.5 |
added 983 characters in body; added 11 characters in body
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| Apr 10, 2010 at 22:50 | comment | added | Jonas Meyer | @Douglas Zare, here's one that was close: mathoverflow.net/revisions/12877/list | |
| Apr 10, 2010 at 22:45 | answer | added | Will Jagy | timeline score: 140 | |
| Apr 10, 2010 at 22:37 | comment | added | Douglas Zare | The edit history is remarkable. Has any question been edited by so many so quickly? I almost joined the list, but I'll hold off since this is about to turn CW... | |
| Apr 10, 2010 at 22:37 | history | edited | Yemon Choi | CC BY-SA 2.5 |
re-edit
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| Apr 10, 2010 at 22:33 | comment | added | Yemon Choi | overlapping edits, it seems. Does anyone object if I put the formulas in equation mode rather than inline? Some of us are making do with small screens and failing eyesight | |
| Apr 10, 2010 at 22:32 | history | edited | Kevin Casto | CC BY-SA 2.5 |
deleted 12 characters in body
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| Apr 10, 2010 at 22:32 | history | edited | Pete L. Clark | CC BY-SA 2.5 |
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| Apr 10, 2010 at 22:32 | history | edited | Yemon Choi | CC BY-SA 2.5 |
fixed the LaTeX
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| Apr 10, 2010 at 22:31 | history | edited | Mariano Suárez-Álvarez | CC BY-SA 2.5 |
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| Apr 10, 2010 at 22:28 | history | asked | Kevin Casto | CC BY-SA 2.5 |