rwg>This pushes the luck frontier back to, e.g., e^-(pi sqrt(3/2)). eta(%e^-(2*sqrt(2)*%pi/sqrt(3))) = Gamma(1/24)*sqrt(sin(%pi/24))*csc(%pi/8)^(1/6) /(2^(23/24)*3^(1/8)*(sqrt(3)-1)^(1/4)*sqrt(%pi)*sqrt(Gamma(1/12))) 2 sqrt(2) %pi - ------------- sqrt(3) eta(%e ) = 1 %pi 1/6 %pi Gamma(--) sqrt(sin(---)) csc (---) 24 24 8 ------------------------------------------------------, 23/24 1/8 1/4 1 2 3 (sqrt(3) - 1) sqrt(%pi) sqrt(Gamma(--)) 12 eta(%e^-(4*sqrt(2)*%pi/sqrt(3))) = (sqrt(3)-1)^(1/8)*Gamma(1/24)*sin(%pi/24)^(1/4) /(2*2^(17/24)*3^(1/8)*sqrt(%pi)*sqrt(Gamma(1/12))*SIN(%PI/8)^(1/12)) 4 sqrt(2) %pi - ------------- sqrt(3) eta(%e ) = 1/8 1 1/4 %pi (sqrt(3) - 1) Gamma(--) sin (---) 24 24 ----------------------------------------------------. 17/24 1/8 1 1/12 %pi 2 2 3 sqrt(%pi) sqrt(Gamma(--)) sin (---) 12 8 (Jacobi's imaginary transformation gives sqrt(6) pi and sqrt(3/2) pi.) All unproven, of course. rwg>So far, no luck finding eta(e^-(pi sqrt(2))) etc, and eta' of anything
besides e^-(2 pi). (One more of the latter would open another floodgate.)
(c665) 'at('diff(eta(q),q,1),q = %e^-(3*%pi)) = -%e^(3*%pi)*thetaderiv[1](%pi/3,2,%e^-(%pi/6))/(24*sqrt(3))) = -2^(1/8)*%e^(3*%pi)*Gamma(1/4)*(Gamma(1/4)^4/(16*%pi^(15/4))-1/%pi^(7/4))/8 | d | (d665) -- (eta(q))| dq | - 3 %pi |q = %e 3 %pi %pi - %pi/6 %e thetaderiv (---, 2, %e ) 1 3 = - -------------------------------------- 24 sqrt(3) 4 1 gamma (-) 1/8 3 %pi 1 4 1 2 %e gamma(-) (---------- - ------) 4 15/4 7/4 16 %pi %pi = - ------------------------------------------- 8 (c666) dfloat(%) | d | (d666) -- (eta(q))| dq | |q = 8.06995175703047d-5 = - 77.8356998979772d0 = - 77.8356998979791d0 So now we can do those log derivative sums, e.g., 'sum(n/(%e^(3*%pi*n)-1),n,1,inf) = 1/24-(2-sqrt(2)*3^(1/4))^(1/4)*((2-sqrt(2)*3^(1/4))^(3/4)*(5*sqrt(2)*3^(3/4)+6*sqrt(3)+9*sqrt(2)*3^(1/4)+6)*Gamma(1/4)^4/(288*(sqrt(3)-1)^(5/12)*%pi^2)+(sqrt(3)-1)^(7/12)/(2-sqrt(2)*3^(1/4))^(1/4))/(12*(sqrt(3)-1)^(7/12)*%pi) inf ==== \ n 1 1/4 1/4
------------- = -- - (2 - sqrt(2) 3 ) / 3 %pi n 24 ==== %e - 1 n = 1
1/4 3/4 3/4 1/4 ((2 - sqrt(2) 3 ) (5 sqrt(2) 3 + 6 sqrt(3) + 9 sqrt(2) 3 7/12 4 1 5/12 2 (sqrt(3) - 1) + 6) gamma (-)/(288 (sqrt(3) - 1) %pi ) + ---------------------) 4 1/4 1/4 (2 - sqrt(2) 3 ) 7/12 /(12 (sqrt(3) - 1) %pi) (I didn't say they'd be nice. The only opportunity here for further simplification is that quadrinomial.) (c658) bfloat(apply_nouns(subst(22,inf,%)),33) (d658) 8.07190569092038104656260759325017b-5 = 8.07190569092038104656260759325078b-5 The e^(2 pi n) case of this is very simple, but I don't see a way of searching for nonhypergeometric series identities on the Wolfram site, nor am I clear whether there are identities there that the latest version of Mma doesn't know. I repeat my plea: Does anybody know where WRI got those valuations of DedekindEta[I] and DedekindEta'[I] that made this all possible? --rwg PS, that Abel-Plana Theta integral further simplifies: 'integrate(sin(y)/((%e^-(2*%pi*y/log(q))-1)*(cos(2*y)+(q^4+1)/(2*q^2))),y,0,inf) = (((q+1)^2/(q^2+1)-theta[3](0,q)^2)*log(q)-4*atan(q))/(4*(1/q-q)) inf / [ sin(y) I ------------------------------------- dy ] 2 %pi y / - ------- 4 0 log(q) q + 1 (%e - 1) (cos(2 y) + ------) 2 2 q 2 (q + 1) 2 (-------- - theta (0, q)) log(q) - 4 atan(q) 2 3 q + 1 = --------------------------------------------. 1 4 (- - q) q (Which we can now give in Gammas for q=e^-(sqrt(rational) pi).)