Jörg>* Bill Gosper <billgosper@gmail.com> [Nov 12. 2012 16:42]: On Wed, Nov 7, 2012 at 3:05 PM, Bill Gosper <billgosper@gmail.com> wrote: [...] In http://oeis.org/A194094 Michael Somos gives: (2/Pi)*elliptic_E(k) = theta_3(q)^2 - 2 * (theta_4(q) / theta_3(q))^2 * Dq ( theta_4(q)^-2 ) = theta_3(q)^2 + 4 Dq (theta_4(q)) / (theta_4(q) * theta_3(q)^2) where Dq (f) := q * df/dq -------- This is presumably d/dq of EllipticK[EllipticTheta[2, 0, q]^4/EllipticTheta[3, 0, q]^4] == 1/2 \[Pi] EllipticTheta[3, 0, q]^2 but it can't be right--there can't be a naked q. I get EllipticE[EllipticTheta[2, 0, q]^4/EllipticTheta[3, 0, q]^4] == (Pi*(-EllipticTheta[2, 0, q]^4 + EllipticTheta[3, 0, q]^4)* Derivative[0, 0, 1][EllipticTheta][2, 0, q])/ (2*EllipticTheta[3, 0, q]*(EllipticTheta[3, 0, q]* Derivative[0, 0, 1][EllipticTheta][2, 0, q] - EllipticTheta[2, 0, q]*Derivative[0, 0, 1][EllipticTheta][3, 0, q])) which is a lot smaller than what I got by converting the 𝜗 to η before differentiating, but has the problem of needing special values for the two 𝜗 and 𝜗'. Can it really be that no one has tabulated EllipticE[E^-(surd π)]? --rwg