Mostly for Neil Bickford, whose pendulum project started this flap, I wrote http://gosper.org/morefou.pdf, with an example and graphs, which almost puts the subject to bed, but at the end opens a new jigger of worms: Can the coefficients in the truncated Fourier series be adjusted to equalize all the ripples, at least when approximating JacobiAM? --rwg Oddity: A few wks ago I reported being thrown by ISC'c failure to find 4 sech(pi/2). Now it works. Although http://functions.wolfram.com/EllipticFunctions/JacobiAmplitude/03/01/ lists only trivialities, JacobiAM has scads of special values. As mentioned in my elliptic K msg, we have scads of special values of eta, from which we can make scads of thetas, from which we can make scads of Jacobi elliptic functions. Rational multiples of the argument u of any of these are just algebraic functions. Other algebraic functions convert everything to sn, and am = arcsin(sn). E.g., JacobiAmplitude[Pi^(3/2)/(6*Gamma[3/4]^2), 1/2] == ArcSin[(1/2)*(1 + Sqrt[2]*3^(1/4) - Sqrt[3])] 3/2 Pi 1 1 1/4 JacobiAmplitude[-----------, -] == ArcSin[- (1 + Sqrt[2] 3 - Sqrt[3])] 3 2 2 2 6 Gamma[-] 4 JacobiAmplitude[(3^(3/4)*(1 + Sqrt[3])*(-Sqrt[2] + Sqrt[3])*Sqrt[Pi]* Gamma[25/24])/(2*Gamma[13/24]), (1/4)*(-1 + Sqrt[3])^4* (Sqrt[2] + Sqrt[3])^2] -> ArcSin[1/Sqrt[1 + Sqrt[1 - (1/4)*(-1 + Sqrt[3])^4*(Sqrt[2] + Sqrt[3])^2]]] 3/4 25 3 (1 + Sqrt[3]) (-Sqrt[2] + Sqrt[3]) Sqrt[Pi] Gamma[--] 24 JacobiAmplitude[----------------------------------------------------------, 13 2 Gamma[--] 24 1 4 2 - (-1 + Sqrt[3]) (Sqrt[2] + Sqrt[3]) ] -> 4 1 ArcSin[----------------------------------------------------------] 1 4 2 Sqrt[1 + Sqrt[1 - - (-1 + Sqrt[3]) (Sqrt[2] + Sqrt[3]) ]] 4 It is probable that these all have u/K(m) rational, which would make them fairly easy to detect and simplify. 4D beer dispenser: http://www.youtube.com/watch?v=wiu_IX14wLI