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Gauss's Transformation

If

(1+xsin^2alpha)sinbeta=(1+x)sinalpha,

then

(1+x)int_0^alpha(dphi)/(sqrt(1-x^2sin^2phi))=int_0^beta(dphi)/(sqrt(1-(4x)/((1+x)^2)sin^2phi)).

See also

Elliptic Integral of the First Kind, Landen's Transformation

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Cite this as:

Weisstein, Eric W. "Gauss's Transformation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GausssTransformation.html

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