Gauss–Legendre algorithm: Difference between revisions

Revision as of 03:37, 12 October 2019

Revision as of 11:11, 12 October 2019

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:Equivalently,

:<math>\forall \varphi: K(\sin\varphi)[E(\cos\varphi)-K(\cos\varphi)] + K(\cos\varphi)E(\sin\varphi) = \frac{\pi}{2}</math>

=== 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>

=== Elementary proof with integral calculus ===