I think this fully characterizes eta (and (q;q)_oo) near the cyclotomic points: Empirically, Here it is again, tested and slightly simplified,
'limit(%e^-(%pi^2/(6*d^2*log(q)))*sqrt(-log(q))*eta(%e^(2*%i*%pi*n/d)*q),q,1) = sqrt(2*%pi/d)*%e^(%i*%pi*g(n/d)/12) 2 i n pi -------- i g(n/d) pi d ----------- sqrt(- log(q)) eta(e q) 12 2 pi limit ------------------------------- = e sqrt(----), q -> 1 2 d pi ----------- 2 6 d log(q) e with a greatly simplified phase fudger g, now with one (rational) argument, more clearly showing the continued fraction recursion, r -> 1/mod(r,1): g(r) := if integerp(r) then r else 3+floor(r)-1/(denom(r)^2*nummod(r,1))-g(1/nummod(r,1)) 1 1 g(r) := if integerp(r) then r else - g(---------) - ------------------- mod(r, 1) 2 denom (r) mod(r, 1) + floor(r) + 3 . --rwg PS, I was startled to see the Merriam Webster dictionary program offer adjective and adverb definitions of "nation". Unable to imagine such usages, I gave up and peeked, expecting to kick myself. I didn't.