I knew that the Arithmetic Geometric Mean had a closed form solution
in terms of K(k) (i.e. the complete elliptic integral of the first kind).
However, recently I stumbled across a statement to the effect that
the AGM also has a solution in terms of E(k)--the complete
elliptical integral of the second kind.

The statement is here:
PlanetMath: arithmetic-geometric mean

To wit: "The AGM can be used to numerically evaluate elliptic
integrals of the first and second kinds."

The website does not offer the closed form in terms of E(k),
however. It only gives the well-known K(k) form.

Is it true that a closed form solution exists for the AGM in terms
of E(k)? If yes, please share!

Thank you