just out of curiosity, is there a point of de-nesting if it leads to more square roots? bob baillie --- Warut Roonguthai wrote:> Another example, sqrt(3+sqrt(5+2*sqrt(3))) => 1/2 + sqrt(3)/2 + sqrt((4-sqrt(3))/2).> > If (a^2-b)^2-c is a perfect square,> then sqrt(a+sqrt(b+sqrt(c))) can be (partially) denested.> > Warut> > > On Sat, Nov 24, 2012 at 4:22 PM, Bill Gosper <billgosper@gmail.com <https://ma.sdf.org/sm/src/compose.php?send_to=billgosper%40gmail.com>> wrote:>> It's clear that √(a+√(b+√c)) can't completely denest unless √(b+√c) does.>> Failing that, I'd've expected nothing better than √(b+√c) + √d + √e + ...,>> i.e., the √(b+√c) would act atomically, like √integer. But>>>> Sqrt[1/2 - Sqrt[-2 + Sqrt[5]]] ==>> 1/2 Sqrt[-1 + Sqrt[5]] + 1/4 (Sqrt[2] - Sqrt[10])>> --rwg First off, denestings (of not just square roots) are interesting, mysterious, pretty, and (Algebra I) accessible. Next, Bruce Berndt once wrote me "In the several years that I devoted to calculating Ramanujan's class invariants and singular moduli, I often needed to denest radicals. Here is a very mild contribution that I made in this direction. www.math.uiuc.edu/~berndt/articles/*radicals*.pdf " Also, denesting sums of radicals can permit additional simplification. What is Sqrt[1/2 - Sqrt[-2 + Sqrt[5]]] - 1/2 Sqrt[-1 + Sqrt[5]] ? Also, denesting radicals inside transcendental functions can permit the invocation of addition formulas. Also, denested expressions are canonical, promoting recognition. Minimal polynomial notation is even more canonical, but conceals essential relationships. But for evaluation, it usually beats both nested and unnested radicals. Finally, denesting can re-express surds with large coefficients as powers of surds with small coefficients: (Sqrt[3] - Sqrt[2])^16 has more square roots than 46099201 - 18819920 Sqrt[6]. Which do you like? --rwg Subj: Re: Boy do we need a good radical simplifier On Thu, Jan 6, 2011 at 11:19 PM, Bill Gosper <billgosper@gmail.com> wrote:
(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