text: Legendre's relation The Legendre's relation or Legendre Identity shows the relation of the integrals K and E of an elliptic modulus and its anti-related counterpart in an integral equation of second degree:
  For two modules that are Pythagorean counterparts to each other, this relation is valid:
  For example:
  And for two modules that are tangential counterparts to each other, the following relationship is valid:
  For example:
  The Legendre's relation for tangential modular counterparts results directly from the Legendre's identity for Pythagorean modular counterparts by using the Landen modular transformation on the Pythagorean counter modulus.
  Special identity for the lemniscatic case
  For the lemniscatic case, the elliptic modulus or specific eccentricity ε is equal to half the square root of two. Legendre's identity for the lemniscatic case can be proved as follows:
  According to the Chain rule these derivatives hold:
  By using the Fundamental theorem of calculus these formulas can be generated:
  The Linear combination of the two now mentioned integrals leads to the following formula:
  By forming the original antiderivative related to x from the function now shown using the Product rule this formula results:
  If the value  is inserted in this integral identity, then the following identity emerges:
  This is how this lemniscatic excerpt from Legendre's identity appears:
  Generalization for the overall case