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AGM / Elliptic integral relationship proof from here: https://en.wikipedia.org/wiki/Arithmetic%E2%80%93geometric_mean#Proof_of_the_integral-form_expression
Mathematics
katieb
  • katieb
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\[I(x,y) = \int\limits_0^{\pi/2} \frac{d \theta}{x^2 \cos^2 \theta + y^2 \sin^2 \theta}\] Is there a geometric reason for the substitution of \[\sin \theta = \frac{ 2x \sin \phi}{(x+y)+(x-y) \sin^2 \phi}\] which allows us to see that: \[I(x,y) = I(\frac{1}{2} (x+y),\sqrt{xy})\]
amoodarya
  • amoodarya
x,y are real number ?
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Yes, I believe in general they are, although I am curious about when x and y are complex.

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Overall I just wanna understand how in the world Gauss was able to come to this relationship between the elliptic integral and AGM honestly haha. The AGM basically is the closed form of an elliptic integral it seems, which I am shocked no one has mentioned this powerful function before.
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I forgot the square root sign here: \[I(x,y) = \int\limits_0^{\pi/2} \frac{d \theta}{\sqrt{x^2 \cos^2 \theta + y^2 \sin^2 \theta}}\]
amoodarya
  • amoodarya
to solve ,we can do \[\int\limits \frac{ d \theta }{ x^2\cos^2\theta+y^2\sin^2\theta }=\\\int\limits \frac{ d \theta }{ 1+\frac{y^2\sin^2\theta}{x^2\cos^2\theta} }\frac{1}{x^2\cos^2\theta}=\\\frac{1}{x^2} \int\limits \frac{ d \theta }{ 1+(\frac{y}{x}\tan \theta)^2 }(1+\tan^2\theta)=\\ u=\frac{y}{x}\tan \theta \\=\frac{1}{x^2}*\frac{x}{y} \arctan(\frac{y}{x}\tan \theta)\]
amoodarya
  • amoodarya
question changed ?!
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Yeah, sorry about that, it wasn't this integral to begin with, I had written it wrong by accident.
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Also, I am not really interested in solving the integral, since it's not solvable.
ganeshie8
  • ganeshie8
You want to show I(a,b) = I(am(a,b), gm(a,b)) using some geometry and relate it to gauss ?
amoodarya
  • amoodarya
yes do you check? \[I(\frac{x+y}{2},\sqrt{xy})\] maybe in writing ,and arranging find the trick !
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Well I just worked it out, but it just seems like a lot of terrible algebra sadly! @amoodarya I guess I would like to see some sort of motivation for why there is any relationship between repeating this process: \[a_{n+1} = \frac{a_n+b_n}{2}\]\[b_{n+1}=\sqrt{a_nb_n}\] and the integral in a way. How did Gauss know to look here or think this? I can follow through the algebra to show that it's true, but I want to know how ellipses and these averages are related, this seems very strange and fascinating!
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It appears as if someone has explained it here, http://paramanands.blogspot.com/2009/08/arithmetic-geometric-mean-of-gauss.html#.VZ0JZPm6fIU I can't seem to follow his reasoning on my own, if anyone wants to try to understand this article with me step by step, speak up! Otherwise I'll just sorta take a break from this haha. I don't even know where elliptic integrals really come from other than in calculating arc length or why they are important at all.

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