Ed Pegg had me snooping in http://functions.wolfram.com/EllipticFunctions/DedekindEta/ where I was royally shocked to find an (unattributed) evaluation of eta(e^(-2 pi)). Which Mma 6.0 knows! This opens the floodgates, permitting closed forms for eta(e^(-r pi)) for *all* positive rational r. E.g., 3/4 1/4 1/6 1 (5 3 - 3 sqrt(2) sqrt(3) - 3 3 ) gamma(-) - 12 %pi 4 [eta(%e ) = -------------------------------------------------, 1/24 3/4 2 2 sqrt(6) %pi 1/6 1 (sqrt(3) - 1) gamma(-) - 6 %pi 4 eta(%e ) = -------------------------, 1/12 3/8 3/4 2 2 3 %pi 1 1 gamma(-) gamma(-) - 4 %pi 4 - 2 %pi 4 eta(%e ) = -------------, eta(%e ) = --------, 3/8 3/4 3/4 2 2 %pi 2 %pi 1 2 %pi gamma(-) - ----- - %pi 4 3 eta(%e ) = -----------, eta(%e ) = 7/8 3/4 2 %pi 1/8 1/6 1 3 (sqrt(3) - 1) gamma(-) 4 - %pi/3 ------------------------------, eta(%e ) = 1/12 3/4 2 2 %pi 3/4 1/4 1/6 1 (5 3 - 3 sqrt(2) sqrt(3) - 3 3 ) gamma(-) 4 -------------------------------------------------] 1/24 3/4 2 2 %pi which in turn provide the corresponding theta[1]'(0,q), and theta[s](0,q): - 3 %pi [theta (0, %e ) = 2 1/6 3/4 1/4 2/3 1 (sqrt(3) - 1) (- 3 - 3 + 3 sqrt(2)) gamma(-) 4 --------------------------------------------------------, 2/3 13/24 3/4 2 2 3 %pi 1 gamma(-) - 3 %pi 4 theta (0, %e ) = ----------------------------------, 3 1/4 3/8 3/4 2 3 sqrt(sqrt(3) - 1) %pi - 3 %pi theta (0, %e ) = 4 5/6 1 (sqrt(3) - 1) gamma(-) 4 ------------------------------------------------] 1/3 5/24 3/4 1/4 2/3 3/4 2 3 (- 3 - 3 + 3 sqrt(2)) %pi There was also an evaluation of eta', giving such goodies as - %pi theta''' (0, %e ) = 1 2 1 %pi - %pi/3 3 1 gamma (-) theta'' (---, %e ) 3 gamma (-) 4 1 3 4 ----------------------------------- = - -----------. 3/2 13/4 2 sqrt(3) %pi 4 %pi (I just noticed %pi 1/3 2 2 2 theta'' (---, q ) eta (q ) 1 3 theta''' (0, q) = ------------------------------.) 1 sqrt(3) Do we agree with Rich that the simplifier should prefer theta''(pi/3) to theta'''(0)? So far, no luck finding eta(e^-(pi sqrt(2))) etc, and eta' of anything besides e^-(2 pi). (One more of the latter would open another floodgate.) Likewise, LatticeReduce found no polynomial for q^(1/24)*(i q;q)_oo for q=e^-pi. --rwg