Gauss–Legendre algorithm: Difference between revisions

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=== Legendre’s identity ===

Legendre proved the following 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(\sincos \varphitheta) E(\sin \theta ) + K(\sin \theta ) E(\sincos \varphitheta) - K(\sincos \varphitheta) K(\sin \theta) = {1\pi \over 2}, \pitext{ for all } \theta. </math><ref name="brent" />

:Equivalently,

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

=== Elementary proof with integral calculus ===