Landen's transformation is a mapping of the parameters of an elliptic integral, useful for the efficient numerical evaluation of elliptic functions. It was originally due to John Landen and independently rediscovered by Carl Friedrich Gauss.
Statement The incomplete elliptic integral of the first kind F is
F ( φ ∖ α ) = F ( φ , sin α ) = ∫ 0 φ d θ 1 − ( sin θ sin α ) 2 , {\displaystyle F(\varphi \setminus \alpha )=F(\varphi ,\sin \alpha )=\int _{0}^{\varphi }{\frac {d\theta }{\sqrt {1-(\sin \theta \sin \alpha )^{2}}}},}
where α {\displaystyle \alpha } is the modular angle. Landen's transformation states that if α 0 {\displaystyle \alpha _{0}} , α 1 {\displaystyle \alpha _{1}} , φ 0 {\displaystyle \varphi _{0}} , φ 1 {\displaystyle \varphi _{1}} are such that ( 1 + sin α 1 ) ( 1 + cos α 0 ) = 2 {\displaystyle (1+\sin \alpha _{1})(1+\cos \alpha _{0})=2} and tan ( φ 1 − φ 0 ) = cos α 0 tan φ 0 {\displaystyle \tan(\varphi _{1}-\varphi _{0})=\cos \alpha _{0}\tan \varphi _{0}} , then
F ( φ 0 ∖ α 0 ) = ( 1 + cos α 0 ) − 1 F ( φ 1 ∖ α 1 ) = 1 2 ( 1 + sin α 1 ) F ( φ 1 ∖ α 1 ) . {\displaystyle {\begin{aligned}F(\varphi _{0}\setminus \alpha _{0})&=(1+\cos \alpha _{0})^{-1}F(\varphi _{1}\setminus \alpha _{1})\\&={\tfrac {1}{2}}(1+\sin \alpha _{1})F(\varphi _{1}\setminus \alpha _{1}).\end{aligned}}}
Landen's transformation can similarly be expressed in terms of the elliptic modulus k = sin α {\displaystyle k=\sin \alpha } and its complement k ′ = cos α {\displaystyle k'=\cos \alpha } .
Complete elliptic integral In Gauss's formulation, the value of the integral
I = ∫ 0 π 2 1 a 2 cos 2 ( θ ) + b 2 sin 2 ( θ ) d θ {\displaystyle I=\int _{0}^{\frac {\pi }{2}}{\frac {1}{\sqrt {a^{2}\cos ^{2}(\theta )+b^{2}\sin ^{2}(\theta )}}}\,d\theta }
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