[math-fun] A cleverness for FullSimplify

FullSimplify[EllipticK[1/2 - 1/2 Sqrt[(-1 + 2 z)^2]] + EllipticK[1/2 (1 + Sqrt[(-1 + 2 z)^2])]] should give EllipticK[1 - z] + EllipticK[z] because it's an even function of sqrt(2z-1), so the branching cancels out. Generalizing slightly, a strange variant of Legendre's relation: EllipticE[1/2 (1 + Sqrt[(-1 + 2 z)^2])] EllipticK[1/2 - 1/2 Sqrt[(-1 + 2 z)^2]] + EllipticE[1/2 - 1/2 Sqrt[(-1 + 2 z)^2]] EllipticK[1/2 (1 + Sqrt[(-1 + 2 z)^2])] - EllipticK[1/2 - 1/2 Sqrt[(-1 + 2 z)^2]] EllipticK[1/2 (1 + Sqrt[(-1 + 2 z)^2])] == Pi/2 Less obviously, (1/2 + 1/(-1 + E^Sqrt[z^2])) Sqrt[z^2] == (1/2 + 1/(-1 + E^z)) z --rwg (MathIsFun)