Trying to assemble an indispensable enhancement theorem

Let $C$ be an aircraft contour offered by the set $\{(x,y) \in \mathbb{R}^2: \, P(x,y) = 0 \}$ and also specify $\omega=\frac{\mathrm{d}x}{y}$. After that, it is

$$\int\limits_0^A \omega + \int\limits_0^B \omega = \int\limits_0^{A \oplus B} \omega$$

(with $\oplus$ being enhancement on a team on the contour ) a theory?

I'm fairly certain this is what's taking place when C is an elliptic contour, yet I have actually fallen short to make this exercise when C is specified by $P(x,y) = x^2 + y^2 - 1$ (device circle ) and also the team regulation is specified by shooting a ray via $(3/5,4/5)$ (picked randomly ) alongside $AB$ and also taking it's junction with the circle as $A \oplus B$.

2019-05-07 10:49:26
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Answers: 2

$0$, the lower restriction for the integrals, need to be whatever factor on the contour is $0$ for the team regulation.

After that $dx/y$ needs to be turning - stable for any kind of circle. It will certainly not hold true for a revolved ellipse ; there is a stable differential yet it is not $dx/y$.

For an elliptic contour, $dx/y$ being the stable differential relies upon the contour being created as $y^2=f(x)$ with $\deg(f)=3$. If you relocate the contour the stable differential will certainly be a various one.

For greater categories the contour does not have an enhancement regulation.

2019-05-09 05:11:53


You're appropriate! This is what's taking place on elliptic curves. And also, if you consider greater categories, as T. claimed, there is no team regulation on the contour. Yet something silly like there not being a team regulation on contours of high category had not been sufficient to stop the 19th century mathematicians, they uncovered that it's the team regulation on an abelian selection pertaining to the contour that matters, and also the contour and also the abelian selection take place to be the very same for cubics!

You can locate all this exercised in any kind of excellent publication that speaks about elliptic integrals or abelian integrals, and also I'm directly a large follower of the presentation in Stillwell's "Mathematics and also its History"

2019-05-08 05:34:09