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Landen's Formula

(theta_3(z,t)theta_4(z,t))/(theta_4(2z,2t))=(theta_3(0,t)theta_4(0,t))/(theta_4(0,2t))=(theta_2(z,t)theta_1(z,t))/(theta_1(2z,2t)),

where theta_i are Jacobi theta functions. This transformation was used by Gauss to show that elliptic integrals could be computed using the arithmetic-geometric mean.

See also

Jacobi Theta Functions

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Cite this as:

Weisstein, Eric W. "Landen's Formula." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LandensFormula.html

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