Timeline for A two-variable Fourier series and a strange integral
Current License: CC BY-SA 2.5
13 events
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| Feb 19, 2010 at 17:09 | comment | added | Kevin Buzzard | Do you know if it's possible to compute values of I(m,n) numerically to, say, 10 dec places? (or even 5 dec places?). Pari's "lindep" function is good at using LLL to spot when a real number is of the form a+b/Pi with a,b rational, so one could try some examples to see whether the conjecture holds out more generally. | |
| Feb 19, 2010 at 16:52 | comment | added | David E Speyer | Oh, wait, you're right and I'm dumb. Now I'm really confused. This is tempting me to actually work out the example in full, but that will take a while. | |
| Feb 19, 2010 at 16:00 | comment | added | Kevin Buzzard | Right. So when you multiply by pi^2 you get a.pi^2+b.pi? What am I doing wrong? | |
| Feb 19, 2010 at 14:37 | comment | added | David E Speyer | David Hansen's conjecture was that the original integral was of the form a+b/pi. | |
| Feb 19, 2010 at 14:18 | comment | added | Kevin Buzzard | @David: "Dropping out the 4 pi^2, we want to show the integrand is of the form a pi +b.". Either I'm misunderstanding or that should be a.pi + b.pi^2? | |
| Feb 18, 2010 at 23:41 | history | edited | David E Speyer | CC BY-SA 2.5 |
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| Feb 18, 2010 at 22:50 | comment | added | Bjorn Poonen | @David: Don't wimp out yet! Your "elliptic curve" is singular at (-1,-1), so it's just a rational curve, which should make your life much, much easier... | |
| Feb 18, 2010 at 22:37 | history | edited | David E Speyer | CC BY-SA 2.5 |
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| Feb 18, 2010 at 22:29 | comment | added | David E Speyer | As for justifying the integration by parts, the right way to do it is to cut out a small disc D around (-1,-1). There is then a completely valid integration by parts, which winds up with a left over term where we integrate over \partial D. One then has to show that this term drops out when we shrink D. But I'll leave that for someone else. | |
| Feb 18, 2010 at 22:28 | comment | added | David E Speyer | Hmmm. You're right that integral z^k w^l/(4+...) diverges at (-1,-1). On the other hand, the particular expressions we get out of integrating by parts look like integral (z-z^{-1}) z^k w^l/(4+...) depending on which variable we integrated on. And that should be convergent. I'll edit to point that out. | |
| Feb 18, 2010 at 22:13 | comment | added | fedja | Well, your integration by parts already seems very suspicious: the final integral you wrote certainly diverges at $(-1,-1)$. Still I like your general idea :). | |
| Feb 18, 2010 at 22:12 | history | edited | David E Speyer | CC BY-SA 2.5 |
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| Feb 18, 2010 at 21:59 | history | answered | David E Speyer | CC BY-SA 2.5 |