(Question from Julian.) Apparently, K(k) has a closed form at and only at "singular values", i.e. when K(1-k)/K(k) = sqrt(rational), (periodic CF). There's no period in the first 10^4 terms of K(2/3)/K(1/3), but, e.g., EllipticK[4 Sqrt[2] (-1 + Sqrt[2])^2] -> ((2 + Sqrt[2]) Pi^(3/2))/(4 Gamma[3/4]^2) EllipticK[(-1 + Sqrt[2])^4] == ((2 + Sqrt[2]) Pi^(3/2))/(8 Gamma[3/4]^2) EllipticK[(-Sqrt[2] + Sqrt[3])^4/(-1 + Sqrt[2])^4] == (3^(3/4) (-1 + Sqrt[2]) (1 + Sqrt[3]) Gamma[1/3]^3)/( 16 2^(5/6) (-Sqrt[2] + Sqrt[3]) Pi) EllipticK[( 16 Sqrt[2] (-Sqrt[2] + Sqrt[3])^2)/((-1 + Sqrt[2])^2 (1 + Sqrt[ 3])^3)] == (3^(1/4) (-1 + Sqrt[2]) (1 + Sqrt[3]) Gamma[1/3]^3)/( 8 2^(5/6) (-Sqrt[2] + Sqrt[3]) Pi) EllipticK[(-1 + 2^(1/4))^4/(1 + 2^(1/4))^4] == ((-1 + Sqrt[ 2])^2 Pi^(3/2))/(8 Sqrt[2] (-1 + 2^(1/4))^2 Gamma[3/4]^2) EllipticK[(8 2^(1/4) (1 + Sqrt[2]))/(1 + 2^(1/4))^4] == ((1 + 2^(1/4))^2 Pi^(3/2))/(2 Sqrt[2] Gamma[3/4]^2) I don't recall seeing the last four (which are equivalent in pairs). I have to check if K(1-k)/K(k) = sqrt(rational1) + i sqrt(rational2) can also work. Also, that unlikely conjecture that denestable <-> factorable into binomials has survived a minor complexes test: DedekindEta[-((I*((I*Pi)/2 - Pi/4))/(2*Pi))] == (Gamma[1/4]*Sqrt[Sqrt[2] + 1]* I^(13/24))/(2^(25/32)*(Sqrt[2] - 1)^(1/8)*Sqrt[2^(1/4)*I - 1]*Pi^(3/4)) (A virtue of binomial factorization is that it tends to minimize the most "deeply rooted" subexpressions, and we know how unwelcome radicalism is around here.) --rwg