Legendre Relation

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

$$E(k)K'(k)+E'(k)K(k)-K(k)K'(k)={\textstyle{1\over 2}}\pi.$$

References

Abramowitz, M. and Stegun, C. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 591, 1972.