Gauss–Legendre algorithm: Difference between revisions

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=== Legendre’s identity ===

Legendre proved the following identity:

:<math display="block">K(\cos \theta) E(\sin \theta ) + K(\sin \theta ) E(\cos \theta) - K(\cos \theta) K(\sin \theta) = {\pi \over 2}, \text{ </math>
for all } <math>\theta. </math><ref name="brent" />.

=== Elementary proof with integral calculus ===