I think this fully characterizes eta (and (q;q)_oo) near the cyclotomic points: Empirically, Here it is again, tested and slightly simplified,
Actually, what I tested were the Mma definitions: HoldForm[Limit][(DedekindEta[n/d - I*Log[q]]*Sqrt[-Log[q]])/ E^(Pi/(12*d^2*Log[q])), q -> 1] == E^((1/12)*I*Pi*g[n/d])/Sqrt[d] n DedekindEta[- - I Log[q]] Sqrt[-Log[q]] 1/12 I Pi g[n/d] d E Limit[---------------------------------------, q -> 1] == ----------------- 2 Sqrt[d] Pi/(12 d Log[q]) E g[r_?ExactNumberQ] := If[IntegerQ[r], r, 3 + Floor[r] - 1/(Denominator[r]^2*Mod[r, 1]) - g[1/Mod[r, 1]]] g[r_?ExactNumberQ] := 1 1 If[IntegerQ[r], r, 3 + Floor[r] - ------------------------- - g[---------]] 2 Mod[r, 1] Denominator[r] Mod[r, 1] where DedekindEta takes argument tau instead of q=e^(2 i pi tau), and has period 24, just like the rhs of the limit, which is good for all relatively prime n and d, d>0. But like Perkin-Elmer, I failed to all-up test after converting back to eta(q) := q^(1/24) (q;q)_oo, which has branch problems. The formula
'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
is good only for 2 |n| < d, n/=0, which is barely enough, with 2 %pi - -------- 6 log(q) eta(q) %e sqrt(- log(q)) -> sqrt(2 %pi), to describe eta(e^(i t pi)) for rational t. A simple example of the eta(q) branch problem: Suppose you try to roll your own in Mma with the handy identity eta[q_] := EllipticTheta[1, Pi/3, q^(1/6)]/Sqrt[3] Now Plot[Arg[{DedekindEta[tau],eta[E^(2*I*Pi*tau)]}],{tau,I/2-3,I/2+3}] (It refuses because Plot needs educating, so mut-mut) and you'll get a wavy ramp and a sawtooth instead of superposed ramps. This is because eta[E^(2*I*Pi*tau)] gives EllipticTheta[1, Pi/3, (E^(2*I*Pi*tau))^(1/6)]/Sqrt[3] instead of EllipticTheta[1, Pi/3, E^((I*Pi*tau)/3)]/Sqrt[3], whose period is 6 times greater. One of those cases where PowerExpand fixes things instead of breaking them: eta[q_] := EllipticTheta[1, Pi/3, PowerExpand[q^(1/6)]]/Sqrt[3] --rwg