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Legendre Relation

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Let E(k) and K(k) be complete elliptic integrals of the first and second kinds, with E^'(k) and K^'(k) the complementary integrals. Then

E(k)K^'(k)+E^'(k)K(k)-K(k)K^'(k)=1/2pi.

See also

Complete Elliptic Integral of the First Kind, Complete Elliptic Integral of the Second Kind

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References

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 591, 1972.Enneper, A. Elliptische Functionen. Halle, Germany: Louis Nebert, 1890.Trott, M. The Mathematica GuideBook for Programming. New York: Springer-Verlag, pp. 64-65, 2004. http://www.mathematicaguidebooks.org/.Trott, M. The Mathematica GuideBook for Symbolics. New York: Springer-Verlag, p. 29, 2006. http://www.mathematicaguidebooks.org/.

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Legendre Relation

Cite this as:

Weisstein, Eric W. "Legendre Relation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LegendreRelation.html

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