Except for positive q. For negative q, empirically, triple 1,2,4 becomes (c67) (ETA(-Q),SUBST((-1)^-(1/6)*%%,%%,SUBST(-Q,Q,D34))); 8 16 4 16 eta (- q) eta (q ) 2/3 16 8 4 24 2 (d67) - ---------------------- - %i eta (- q) eta (q ) = eta (q ) 2/3 %i I.e., we need to multiply eta(-q) by e^(-i pi/6) in the usual aequatio, which was d34. Testing numerically, (c68) EXPAND(BFLOAT(SUBST(69/105,Q,%))); (d68) 6.1178890272559794661b-22 %i + 1.1762322592495258282b-10 = 1.1762322592637729936b-10 Testing symbolically, (c64) (ETA(-%E^-%PI),SUBST((-1)^-(1/6)*%%,%%,D56)); 2/3 8 - 4 %pi 16 - %pi (d64) - %i eta (%e ) eta (- %e ) 16 - 4 %pi 8 - %pi 16 eta (%e ) eta (- %e ) 24 - 2 %pi - ----------------------------------- = eta (%e ) 2/3 %i (c66) SUBST([D46,D48,D50],D64); 24 1 24 1 gamma (-) gamma (-) 4 4 (d66) -------------- = -------------- 18 18 16777216 %pi 16777216 %pi where (c69) [D45,D46,D48,D50]; 1 1 gamma(-) gamma(-) - %pi 4 - 2 %pi 4 (d69) [eta(%e ) = -----------, eta(%e ) = --------, 7/8 3/4 3/4 2 %pi 2 %pi 1 1/4 1/12 1 gamma(-) 2 %i gamma(-) - 4 %pi 4 - %pi 4 eta(%e ) = -------------, eta(- %e ) = --------------------] 3/8 3/4 3/4 2 2 %pi 2 %pi The obvious suggestion is that eta(-q) is off by i^(-1/3), both numerically and symbolically. But here are four different expressions for eta(-e^-pi): Out[100]= {((-1)^(1/24)*Gamma[1/4])/(2*Pi)^(3/4), (-1)^(1/24 + I/24)* QPochhammer[-E^(-Pi), -E^(-Pi)], DedekindEta[1/2 + I/2], EllipticTheta[1, Pi/3, (-1)^(1/6 + I/6)]/Sqrt[3]} In[101]:= N[%] Out[101]= {0.9057633419831963 + 0.11924600619519435*I, 0.9057633419831964 + 0.11924600619519436*I, 0.9057633419831964 + 0.11924600619519445*I, 0.9057633419831963 + 0.11924600619519422*I} which seem to refute the i^(-1/3) theory. Could eta be defined wrong? --rwg