Gauss–Legendre algorithm: Difference between revisions

Revision as of 19:58, 10 June 2023

Revision as of 14:49, 26 November 2023

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where <math>E(k)</math> is the [[Elliptic integral#Complete elliptic integral of the second kind|complete elliptic integral of the second kind]]:

:<math>E(k) = \int_0^{\pi/2}\sqrt {1-k^2 \sin^2\theta}\; d\theta</math> and <math>K(k) = \int_0^{\pi/2}\frac{d\theta}{\sqrt {1-k^2 \sin^2\theta}}.</math>

and

Gauss knew of both of these results.<ref name="brent">{{Citation

:<math>K(k) = \int_0^{\pi/2}\frac{d\theta}{\sqrt {1-k^2 \sin^2\theta}}.</math>

Gauss knew of boththese of thesetwo results.<ref name="brent">{{Citation

| last=Brent

| first=Richard

Line 104 ⟶ 108:

| accessdate=

}}</ref>

=== Legendre’s identity ===

Legendre proved the following identity: