(Apologies if my suspicion of having already sent this is correct.) http://en.wikipedia.org/wiki/Theta_function gives several slightly unsimplified valuations of theta_3(0,e^-(n pi)), and then EllipticTheta[3, 0, E^-(6*\[Pi])] -> 3^(5/8)*(3*Sqrt[2] + 3^(5/4) + 2*Sqrt[3] - 3^(3/4) + 12^(3/4) - 4)^(1/3)/6/(1 + Sqrt[6] - Sqrt[2] - Sqrt[3])^(1/6)*\[Pi]^(1/4)/ Gamma[3/4] a cube root over a 6th root instead of just a square root: (Sqrt[1+Sqrt[2]+3^(1/4)+Sqrt[3]]*Pi^(1/4))/(2^(3/4)*3^(3/8)*Gamma[3/4]) i.e. 1/4 1/4 -6 Pi Sqrt[1 + Sqrt[2] + 3 + Sqrt[3]] Pi EllipticTheta[3, 0, E ] -> ---------------------------------------- 3/4 3/8 3 2 3 Gamma[-] 4 This is a respectable approximation to 1: N[%, 22] 1.000000013024824272160 -> 1.000000013024824272160 Mathematica has the requisite machinery, but requires constant manual struggling against its thuggish nesting of radicals and reversion to Root notation. It needs to distinguish Radical Number from Algebraic Number. It also needs some way to decache generated polynomials, which seem to be causing a storage leak. --rwg