rwg>I repeat my plea: Does anybody know where WRI got those valuations of
DedekindEta[I] and DedekindEta'[I] that made this all possible?
Well, just the former, which Jacobi imaginary-transforms to itself. But if you differentiate the transformation before plugging in tau = I, you immediately get | 2 %pi - 2 %pi d | %e eta(%e ) -- (eta(q))| = ---------------------- dq | - 2 %pi 4 %pi |q = %e | d | - -- (eta(q))| dq | - 2 %pi |q = %e
(I didn't say they'd be nice. [...] But some of them are: 'sum(n/(%e^(6*%pi*n)-1),n,1,inf) = -sqrt(2)*3^(3/4)*Gamma(1/4)^4/(864*(sqrt(3)-1)*%pi^3)-1/(24*%pi)+1/24
inf 3/4 4 1 ==== sqrt(2) 3 Gamma (-) \ n 4 1 1
------------- = - ---------------------- - ------ + -- / 6 %pi n 3 24 %pi 24 ==== %e - 1 864 (sqrt(3) - 1) %pi n = 1
Taking only four terms of the sum, (c848) bfloat(apply_nouns(subst(4,inf,%)),32) (d848) 6.5124122633144373234994796373366b-9 = 6.512412263314437323499479556926b-9 Notice how the rhs approximates 0. Can its baby brother be new? 'Sum(n/(%e^(%pi*n)-1),n,1,inf) = Gamma(1/4)^4/(64*%pi^3)-1/(4*%pi)+1/24 inf 4 1 ==== gamma (-) \ n 4 1 1 > ----------- = --------- - ----- + -- / %pi n 3 4 %pi 24 ==== %e - 1 64 %pi n = 1 Still open: eta'(e^-(pi sqrt r)), and thus the corresponding sums like the above. --rwg