rwg> Actually, we need 2 |n| <= d, which fortunately obtains.
Actually just -d < 2 n <= d. The limit formula works for all (reduced) n/d with the tau flavor of eta (period 24), but the q = e^(2 i pi tau) flavor introduces this artificial period of 1. Those of you deterred by Macsyma's ugly ASCII displays of my recent eta evaluations are invited to browse http://gosper.org/etavals.html . It's already obsolete, and will likely get updated. Note that many of the special values of thetas can be written as etas, and thus we now have special values of theta derivatives, e.g., pi - ------- sqrt(3) (d105) theta''(0, e ) = 3 1/3 6 1 2 sqrt(3) Gamma (-) 7/8 3/2 1 3 3 Gamma (-) (---------------------- + 1) 3 3 32 pi - --------------------------------------------- 2/3 2 2 pi (c106) dfloat(%) (d106) - 1.32688190926034d0 = - 1.32688190926035d0 (Despite it's simple appearance, this was a tedious derivation, but I think the process can be automated, especially if nasty algebraic expressions regularly disappear as they did here. The automation of the underlying eta' evaluations is a little iffy. They're really just numerical conjectures.) --rwg