How can the arithmetic-geometric mean be derived using Elliptic Integration?

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yanshu
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hello every,how to get the arithmetic-geometric mean? sorry,i can't find it in other place.and my english is poor. arithmetic-geometric mean is: the limit of An,A1=a,B1=b,A(n+1)=(An+Bn)/2,B(n+1)=(AnBn)^1/2 GAUSS got it. a form of Elliptic Integration. Thanks~:biggrin:
 

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there isn't the proof... how to get it to form of Elliptic Integration?