OOPTH, I was working from an old, cached EllipticLambdaFunction.html. It appears the 17th singular value has suffered a gravitational singularity. Unfortunately, the 11th seems broken also, and the denester and I can't find a correct value as elegant: In[211]:=LambdaStar[11]== Sqrt[(1/12)*(6-Sqrt[3*(9-2*(-9+Sqrt[33])*(17+3*Sqrt[33])^(1/3)+ (27-5*Sqrt[33])*(17+3*Sqrt[33])^(2/3))])] Eric could use a fact checker, although he might not want my values, which are mere empiricisms. But he could have been nice and saved me the trouble of deriving LambdaStar[r]^2+LambdaStar[1/r]^2==1 . Neither is ModularLambda[I*x] + ModularLambda[I/x] == 1 mentioned in Properties & Relations of the Mma ModularLambda doc. --rwg On Sun, Feb 13, 2011 at 2:05 AM, Bill Gosper <billgosper@gmail.com> wrote:
For the 17th singular value, Mathworld gives a nonsensical surdoma with a chunky subexpression both added and subtracted. I PSLQed for it a big bioctic and manually massaged it down to LambdaStar[17]==1/8 (3 + Sqrt[17]) (2 Sqrt[2] - Sqrt[1 + Sqrt[17]]) - Sqrt[-19 Sqrt[2] - 5 Sqrt[34] + Sqrt[1 + Sqrt[17]] (13 + 3 Sqrt[17])]/(2 2^(1/4)) which the denester considers finished. --rwg Does anybody know if the sum of two (or more) undenestable surds can denest? (Nontrivially)
On Sun, Feb 13, 2011 at 12:23 AM, Bill Gosper <billgosper@gmail.com>wrote:
For lamba*[4/5] Mathworld gives a dense reciprocal octic. (Antoctic??) Sqrt[ModularLambda[(2 I)/Sqrt[5]]] == 35 + 25 Sqrt[2] + 16 Sqrt[5] + 11 Sqrt[10] + Sqrt[38 + 17 Sqrt[5]] (-8 + Sqrt[2] - 3 Sqrt[10]) --rwg
On Sun, Feb 13, 2011 at 12:04 AM, Bill Gosper <billgosper@gmail.com>wrote:
Mathworld gives as an unsolved biquartic. LambdaStar[3/4]=2 2^(1/4) Sqrt[-147 - 104 Sqrt[2] + 85 Sqrt[3] + 60 Sqrt[6]] --rwg