For a horizontally swung pendulum (thetamax = pi/2), the Fourier series for am gives instead a fundamental of 4 sech(pi/2) = 1.594, exceeding pi/2 by < 2%. The "sechand" pi/2 fell out of some EllipticKs whose values were among the few currently known to FunctionExpand. But my more-or-less systematic collection of special values of eta provides an abundance of K values, because EllipticK[EllipticTheta[2, 0, q]^4/EllipticTheta[3, 0, q]^4] -> 1/2 Pi EllipticTheta[3, 0, q]^2 , i.e., EllipticK[(16 EllipticEta[q]^8 EllipticEta[q^4]^16)/EllipticEta[q^2]^24] -> Pi EllipticEta[q^2]^10/(2 EllipticEta[q]^4 EllipticEta[q^4]^4) , where EllipticEta[q_] -> EllipticTheta[1, \[Pi]/3, q^(1/6)]/Sqrt[3] , I.e., DedekindEta[tau]->EllipticEta[Exp[2*I*Pi*tau]] E.g., EllipticK[1/2 + Sqrt[2] 3^(1/4) (-2 + Sqrt[3])] -> Gamma[1/4]^2/(2 3^(3/4) (-1 + Sqrt[3]) Sqrt[2 Pi]), EllipticK[1/2 - Sqrt[2] 3^(1/4) (-2 + Sqrt[3])] -> 3^(1/4) Gamma[1/4]^2/(2 (-1 + Sqrt[3]) Sqrt[2 Pi]), EllipticK[1 - (-2 + Sqrt[2] - Sqrt[3] + Sqrt[6])^4] -> 3^(3/4) Gamma[1/3]^3)/ (2 2^(5/6) (1 + Sqrt[3])^(5/2) Sqrt[-3 - 4 Sqrt[2] + 5 Sqrt[3]] Pi) EllipticK[8 2^(1/4) (9 - 4 2^(1/4) - 3 Sqrt[2]))/(-1 + Sqrt[2])^6] -> (-1 + Sqrt[2])^(5/2) Gamma[1/4]^2/(4 Sqrt[2 (9 - 4 2^(1/4) - 3 Sqrt[2]) Pi]) EllipticK[-16 + 12 Sqrt[2]] -> 2 (2 + Sqrt[2]) Gamma[5/4]^2)/Sqrt[Pi] --rwg