On 2015-08-12 01:29, Bill Gosper wrote:
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
Foo, this is just Legendre's relation with k = 1/2 - 1/2 Sqrt[(-1 + 2 z)^2] (We could as well have k=sin^2, k'=cos^2, or sech^2 and tanh^2.)
Less obviously, (1/2 + 1/(-1 + E^Sqrt[z^2])) Sqrt[z^2] == (1/2 + 1/(-1 + E^z)) z
--rwg (MathIsFun)