Timeline for To integrate elliptic integral, we glue two Riemann surface to make torus
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| May 31, 2021 at 15:29 | comment | added | Duality | @Alexandre Eremenko :The integrand is single valued over torus, right? | |
| May 31, 2021 at 14:58 | comment | added | Alexandre Eremenko | @Nekojiru: No, I mean what I wrote: integrand, that is the differential which stands under the integral sign. The general modern tendency is to study these differentials by themselves, rather than integrals. They are called Abelian differentials. | |
| May 31, 2021 at 8:06 | comment | added | Duality | @Alexandre Eremenko You mean, our goal is to make integral well defined. The first attempt is to make integrand single valued ( in this process we make torus), and the second process can be achieved by dividing by period. | |
| May 31, 2021 at 6:46 | comment | added | Duality | @Alexandre Eremenko Integrand is not single valued on torus? The integral depends on the paths, but the integrand itself maybe single valued ( we glued two Riemann surface to make it single valued). What is wrong ? | |
| May 30, 2021 at 6:58 | vote | accept | Duality | ||
| May 31, 2021 at 6:46 | |||||
| May 29, 2021 at 14:27 | comment | added | user21167 | So basic reason is that the curve defined by the equation $X=\{y^2=(1-t^2)(1-k^2t^2)\}\subset {\mathbb CP}^2$ is a torus topologically. We can understand it in another way: the rieman surface of the function $y=\sqrt{(1-t^2)(1-k^2t^2)}$ is a torus. So although the integral of our differential form along some loop may be non-zero, function itself does not change when we follow round this loop. | |
| May 29, 2021 at 14:09 | comment | added | Alexandre Eremenko | @Nekorju: The torus is necessary to make the integrand single valued. The integral is still multi-valued, but its multi-valuedness of the simple kind: it is additive. | |
| May 29, 2021 at 7:53 | comment | added | abx | We do this because an elliptic integral is a holo- (or mero-) morphic form on the torus, not on the sphere. What you call"indeterminacy by paths" is actually a great asset, giving rise to the theorems of Abel and Jacobi. | |
| May 29, 2021 at 4:40 | review | Low quality posts | |||
| May 29, 2021 at 7:26 | |||||
| May 29, 2021 at 3:35 | comment | added | Duality | I'm asking the necessity of making torus. Why can't we deal with the integral on RIeman sphere? Maybe, The function we integrate is multi valued, so we avoid the multi valueness by making torus. But indeterminacy by paths is still unsolved. But the main point to make torus is to avoid multi values by making torus, right ? | |
| May 28, 2021 at 19:30 | history | answered | Alexandre Eremenko | CC BY-SA 4.0 |