We write E_k for \Eta(x^k). One more expression for \Eta(i*x):
# # [ 16 ] # | E | # 1 | 4 |1/4 4 2 2 # \Eta(x) = -- | --- - 4 x U | \where U = E E E # E [ U ] 8 4 2 # 4 #
Expressions for \Eta(W*x) where W is the third root of unity are
# # [ R 3 ]1/3 # \Eta(x) = | -- - 3 x E | \where # | E 9 | # [ 9 ] # # [ 3 12 12 ]1/3 # R = | 27 x E + E | # [ 9 3 ] #
Using the conventional eta with q^(1/24) and tweaking your branches, ETA(%E^(2*%I*%PI/3)*Q) = (%E^(%I*%PI/12)*(27*ETA(Q^9)^12+ETA(Q^3)^12)^(1/3)/ETA(Q^9)+3*ETA(Q^9)^3/SQRT(%I))^(1/3) 2 i pi ------ 3 eta(%e q) = i pi ---- 12 12 9 12 3 1/3 3 9 %e (27 eta (q ) + eta (q )) 3 eta (q ) 1/3 (------------------------------------ + ----------) 9 sqrt(i) eta(q ) has a difference plot qualitatively similar to my eta(i q) formula, except that the wedge is |carg| < pi/9 and the fronds on the central sprout curl promptly out of the zero plane, leaving just a disk on a stem. Oops, except maybe the tiny 9th(?) pair, which seem(s) to be on the level! A super zoom might reveal others. I'm not even sure if the number of fronds is finite. --rwg
and
# # [ 1 / 4 3 3 4 \ ]1/3 # \Eta(x) = | -- | E + 9 x E E - 3 x E | | # | E \ 3 3 27 9 / | # [ 9 ] #
These follow from relations given in Somos' eta07.gp
The following clean relations for fifth and seventh roots do not seem to be solvable (for E1):
E1^5*E25 +5*x*E1^4*E25^2 +15*x^2*E1^3*E25^3 +25*x^3*E1^2*E25^4 \ +25*x^4*E1*E25^5 -1*E5^6
E1^7*E49 +147*x^8*E1^3*E49^5 +21*x^4*E1^5*E49^3 +343*x^10*E1^2*E49^6 \ +343*x^12*E1*E49^7 +35*x^3*E1^2*E7^4*E49^2 +49*x^5*E1*E7^4*E49^3 \ +49*x^6*E1^4*E49^4 +7*x*E1^3*E7^4*E49 +7*x^2*E1^6*E49^2 -1*E7^8
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