[math-fun] Boy do we need a good radical simplifier
(Apologies if my suspicion of having already sent this is correct.) http://en.wikipedia.org/wiki/Theta_function gives several slightly unsimplified valuations of theta_3(0,e^-(n pi)), and then EllipticTheta[3, 0, E^-(6*\[Pi])] -> 3^(5/8)*(3*Sqrt[2] + 3^(5/4) + 2*Sqrt[3] - 3^(3/4) + 12^(3/4) - 4)^(1/3)/6/(1 + Sqrt[6] - Sqrt[2] - Sqrt[3])^(1/6)*\[Pi]^(1/4)/ Gamma[3/4] a cube root over a 6th root instead of just a square root: (Sqrt[1+Sqrt[2]+3^(1/4)+Sqrt[3]]*Pi^(1/4))/(2^(3/4)*3^(3/8)*Gamma[3/4]) i.e. 1/4 1/4 -6 Pi Sqrt[1 + Sqrt[2] + 3 + Sqrt[3]] Pi EllipticTheta[3, 0, E ] -> ---------------------------------------- 3/4 3/8 3 2 3 Gamma[-] 4 This is a respectable approximation to 1: N[%, 22] 1.000000013024824272160 -> 1.000000013024824272160 Mathematica has the requisite machinery, but requires constant manual struggling against its thuggish nesting of radicals and reversion to Root notation. It needs to distinguish Radical Number from Algebraic Number. It also needs some way to decache generated polynomials, which seem to be causing a storage leak. --rwg
Condensing a published value of Theta_4(0,-e^(-6 pi)), Corey's nascent denester uncovered this curious simplification: (1 + Sqrt[3] + Sqrt[2] 3^(3/4))^(1/3)/(Sqrt[3]-1)^(1/6) -> (1 + Sqrt[3] + Sqrt[2] 3^(1/4))^(1/2)/2^(1/6) It also reduces the nesting depth of LambdaStar[5] := Sqrt[ModularLambda[Sqrt[-5]]]: In[249]:= DenestRadicals[Sqrt[1/2 - Sqrt[Sqrt[5] - 2]]] [...] In[250]:= ToRadicals[%] Out[250]= (1 - Sqrt[5] + 3 Sqrt[-2 + Sqrt[5]] + (-2 + Sqrt[5])^(3/2))/(2 Sqrt[2]) which canNOT be done by working outward from the innermost. --rwg
* Bill Gosper <billgosper@gmail.com> [Feb 11. 2011 09:45]:
[...] Out[250]= (1 - Sqrt[5] + 3 Sqrt[-2 + Sqrt[5]] + (-2 + Sqrt[5])^(3/2))/(2 Sqrt[2])
which canNOT be done by working outward from the innermost.
Your (1 - sqrt(5) + 3*sqrt(-2+sqrt(5)) + (-2+sqrt(5))^(3/2))/(2*sqrt(2)) = 1/2*(sqrt(sqrt(5)-1) - sqrt(3-sqrt(5))) = 0.1188769458... (sqrt of 5th singular value)
--rwg [...]
in mathematica 8: c = (1 + Sqrt[3] + Sqrt[2] 3^(3/4))^(1/3)/(Sqrt[3] - 1)^(1/6); FullSimplify[c] = (2*(7 + 3*Sqrt[3] + Sqrt[72 + 42*Sqrt[3]]))^(1/6) which has even fewer radicals than the one given just below. of course, one can always ask, what does "simplest" mean? bob --- Bill Gosper wrote:
Condensing a published value of Theta_4(0,-e^(-6 pi)), Corey's nascent denester uncovered this curious simplification: (1 + Sqrt[3] + Sqrt[2] 3^(3/4))^(1/3)/(Sqrt[3]-1)^(1/6) -> (1 + Sqrt[3] + Sqrt[2] 3^(1/4))^(1/2)/2^(1/6)
It also reduces the nesting depth of LambdaStar[5] := Sqrt[ModularLambda[Sqrt[-5]]]:
In[249]:= DenestRadicals[Sqrt[1/2 - Sqrt[Sqrt[5] - 2]]] [...] In[250]:= ToRadicals[%]
Out[250]= (1 - Sqrt[5] + 3 Sqrt[-2 + Sqrt[5]] + (-2 + Sqrt[5])^(3/2))/(2 Sqrt[2])
which canNOT be done by working outward from the innermost. --rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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Bill Gosper -
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Robert Baillie