fxtbook: "See Mathworld." Mathworld gives a bogus negative value. Not in DLMF? Censored from https://en.wikipedia.org/wiki/User:Singular_Modulus functions.wolfram.com gives Out[839]= ( 32 3^(2/3) + 3 (9 + 7 Sqrt[33])^(1/3) - 4 3^(1/3) (9 + 7 Sqrt[33])^( 2/3))/(4 (6 (9 + 7 Sqrt[33])^(1/3) + Sqrt[ 3 (-32 3^(2/3) (9 + 7 Sqrt[33])^(1/3) + 9 (9 + 7 Sqrt[33])^(2/3) + 4 3^(1/3) (9 + 7 Sqrt[33]))])) In[841]:= (EllipticK[1 - %%]/EllipticK[%%])^2 // N Out[841]= 11. Out[842]= 1/2 + Sqrt[11]/12 - (-21 Sqrt[3] + 11 Sqrt[11])^(1/3)/6 - (21 Sqrt[3] + 11 Sqrt[11])^(1/3)/6 In[843]:= (EllipticK[1 - %]/EllipticK[%])^2 // N Out[843]= 11. --rwg -------- Original Message -------- Subject: [math-fun] the 19th singular value Date: 2011-02-18 01:52 From: Bill Gosper <billgosper@gmail.com> To: math-fun@mailman.xmission.com Reply-To: math-fun <math-fun@mailman.xmission.com> ETA(%E^-(2*SQRT(19)*%PI)) = ((3*SQRT(57)+1)^(1/3)-8/(3*SQRT(57)+1)^(1/3))^(1/8)*(GAMMA(1/38)*GAMMA(5/38)*GAMMA(9/38)*GAMMA(11/38)*GAMMA(7/19)*GAMMA(17/38)*GAMMA(23/38)*GAMMA(25/38)/(38*GAMMA(13/38)*GAMMA(8/19)))^(1/4)/(2*2^(53/152)*%PI) where Corey's denester neatly whacks the 8th root: (-(8/(1 + 3*Sqrt[57])^(1/3)) + (1 + 3*Sqrt[57])^(1/3))^(1/8) == -(2^(3/8)/3) + ((-1 + 3*Sqrt[57])^(1/3)*(4*(-5 + Sqrt[57]) + (11 + Sqrt[57])*(-1 + 3*Sqrt[57])^(1/3)))/(96*2^(5/8)) . It also denests LambdaStar[19], but the result is badly undersimplified. Fullsimplify thuggishly reverts to the original Root object, even when given a ComplexityFunction that enormously penalizes _Root ! But it is possible to guess the general form from the denester mess and determine coefficients with PSLQ: LambdaStar[19]==(((-Sqrt[57] + Sqrt[19] + 7*Sqrt[3] + 11)*(3*Sqrt[57] + 1)^(1/3)/ 16) - ((Sqrt[57] + Sqrt[19] - 7*Sqrt[3] + 11)/(2*(3*Sqrt[57] + 1)^(1/3))) - ((Sqrt[19] - 1)/2))/(3*Sqrt[2]) I was unable to Google whether this denesting is known. Is there a table of singular values more extensive than MathWorld's? --rwg Joerg>Let's not forget Broadhurst's very neat expressions: Message-ID: <200803140250.m2E2oHi4002315@philter.princeton.idaccr.org> Date: Thu, 13 Mar 2008 22:50:17 -0400 From: David Broadhurst <D.Broadhurst@open.ac.uk> To: NMBRTHRY@LISTSERV.NODAK.EDU Reply-To: David Broadhurst <D.Broadhurst@open.ac.uk> Subject: Singular value for Euler's numerus idoneus N=1365 default(realprecision,1050); a_1 = 550*sqrt(13) + 318*sqrt(39) + 750*sqrt(7) + 433*sqrt(21); a_2 = 1986 + 1145*sqrt(3) + 208*sqrt(91) + 120*sqrt(273); b = ((a_1-a_2)^2 - 2)/2; k_1365 = sqrt(1/2 - b^2/4*sqrt(4-b^2) - (2-b^2)/4*sqrt(1-b^2)); --------------- The denester also reduced the depth of k_1365, but made a proverbial Very Large Expression, more than Mathematica could further chew. _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun