In the case of the circle I can hardly make any conclusions from the integral $(1)$, most of the theorems come from geometrical considerations. It's not clear how to prove periodicity from this integral or derive the addition theorem.


$$\arcsin(y) = \displaystyle \int\_{0}^{y} \frac{\mathrm dy}{\sqrt{1 - y^2}} (1)$$


In the case of the lemniscate $(2)$ we (Fagnano) can derive a doubling theorem by trying out substitutions in analogy with those which rationalize the first integral. (I learned this from Siegel - Topics in Complex Function Theory). One result (of the sort I wish I could find more) which does come nicely from this integral is via the substitution $y \mapsto iy$ we notice it has double periodicity.


$$\text{sl}^{-1}(y) = \displaystyle \int\_{0}^{y} \frac{\mathrm dy}{\sqrt{1 - y^4}} (2)$$


Euler and others were able to produce theorems about the elliptic integral $(3)$ by analogy with the lemniscate (I read this in Stillwell - Mathematics and its History) - Just as ideas from $\arcsin$ helped to produce theorems about the lemniscate integral. Still, these theorems are very hard earned and it appears that you have to be a master like Euler to derive them.


$$F(y,k) = \displaystyle\int\_0^y\frac{\mathrm dy}{\sqrt{(1-k^2 y^2)(1-y^2)}} (3)$$




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What I would really like to know is, can we derive more results about the functions from these integrals - maybe using integral manipulation techniques I just don't know about?


Another question that has been bothering me deeply is the integrands (which I believe are called invariant differentials of the lie groups for the algebraic curve) - What sort of coincidence is it that allows the integrands to be of this particular form?


Thank you!



What is the advantage of inverting elliptic integrals?
Some (self biased) links to get you started:


**Expository:**


[Ziegler - talk on the AGM at the Technion's math-club](http://www.technion.ac.il/~tamarzr/agm-talk.pdf)


[Cox: The arithmetic-geometric mean of Gauss.. Enseign. Math. (2) 30 (1984), no. 3-4.](http://retro.seals.ch/cntmng?type=pdf&rid=ensmat-001%3A1984%3A30%3A%3A87&subp=hires)


[Ritzenthaler: AGM for elliptic curves.](http://iml.univ-mrs.fr/~ritzenth/cours/AGM-ec.pdf)


**Textbook:**


[Borwein & Borwein: Pi and the AGM via Amazon (w/ look inside)](http://rads.stackoverflow.com/amzn/click/047131515X)


**Research:** 


[Donagi & Livne:The arithmetic-geometric mean and isogenies for curves of higher genus](http://arxiv.org/abs/alg-geom/9712027)


[Dolgachev & Lehavi: On isogenous principally polarized abelian surfaces](http://arxiv.org/abs/0710.1298)


[Lehavi & Ritzenthaler: Formulas for the arithmetic geometric mean of curves of genus 3](http://arxiv.org/abs/math/0403182)


[Humbert: Sur les fonctions abéliennes singulières (Troisième Mémoire)](http://math-tenere.ujf-grenoble.fr/JMPA/feuilleter.php?id=JMPA_1901_5_7) 




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And a couple of non online sources:


Richelot, De transformatione integralium Abelianorum primi ordinis comentatio.
J. reine angew. Math. 16 (1837) 221–341.


J.-B. Bost and J.-F. Mestre, Moyenne arithmético-géométrique et périodes des courbes de genre 1 et 2, Gaz. Math. 38 (1988), 36–64.


