Legendre's relation

The two forms are equivalent as the periods and quasiperiods can be expressed in terms of complete elliptic integrals.

This form of Legendre's relation expresses the fact that the Wronskian of the complete elliptic integrals (considered as solutions of a differential equation) is a constant.

This relation can be proved by integrating the Weierstrass zeta function about the boundary of a fundamental region and applying Cauchy's residue theorem.

Proof of the derivative of the elliptic integral of the first kind: Proof of the derivative of the elliptic integral of the second kind: For the Pythagorean counter-modules and according to the chain rule this relation is valid: Because the derivative of the circle function is the negative product of the so called identical function and the reciprocal of the circle function.

The Legendre's relation always includes products of two complete elliptic integrals.