Let and be complete elliptic integrals of the first and second kinds, with and the complementary integrals. Then
Legendre Relation
See also
Complete Elliptic Integral of the First Kind, Complete Elliptic Integral of the Second KindExplore with Wolfram|Alpha
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/.Referenced on Wolfram|Alpha
Legendre RelationCite this as:
Weisstein, Eric W. "Legendre Relation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/LegendreRelation.html