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which both converge to the same limit.<br />
If <math>a_0=1\!</math> and <math>b_0=\cos\varphi\!</math> then the limit is <math display="inline">{\pi \over 2K(\sin\varphi)}\!</math> where <math>K(k)\!</math> is the [[Elliptic integral#Complete elliptic integral of the first kind|complete elliptic integral of the first kind]]
:<math>K(k) = \int_0^{\pi/2} \frac{d\theta}{\sqrt{1-k^2 \sin^2\theta}}.\!</math>
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=== Legendre’s identity ===
For <math>\varphi\!</math> and <math>\theta\!</math> such that <math display="inline">\varphi+\theta={1 \over 2}\pi\!</math> Legendre proved the identity:
:<math>K(\sin \varphi) E(\sin \theta ) + K(\sin \theta ) E(\sin \varphi) - K(\sin \varphi) K(\sin \theta) = {1 \over 2}\pi.\!</math><ref name="brent" />
=== Gauss–Euler method ===
The values <math display="inline">\varphi=\theta={\pi\over 4}\!</math> can be substituted into Legendre’s identity and the approximations to K, E can be found by terms in the sequences for the arithmetic geometric mean with <math>a_0=1\!</math> and <math display="inline">b_0=\sin{\pi \over 4}=\frac{1}{\sqrt{2}}\!</math>.<ref>Adlaj, Semjon, ''An eloquent formula for the perimeter of an ellipse'', Notices of the AMS 59(8), p. 1096</ref>
== See also ==
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