On 12/16/10, Bill Gosper <billgosper@gmail.com> wrote:
Simplifying with the aequatio abstrusa: EllipticK[1/(EllipticEta[q]^8/(16*EllipticEta[q^4]^8) + 1)] -> Pi*EllipticEta[q^2]^10/ (2*EllipticEta[q]^4*EllipticEta[q^4]^4) E.g., EllipticK[(2 + GoldenRatio^(3/2))/ (Sqrt[2 + Sqrt[5]] + GoldenRatio^(3/2))] -> (5/2)^(3/4) (5 + Sqrt[5])^(1/4) Gamma[9/20] Gamma[21/20]/Sqrt[Pi]
DAWK! Rich points out this lhs is just EllipticK[1/2 + GoldenRatio^-(3/2)] . Now that's almost unexotic! Betcha EllipticK[1/2 - GoldenRatio^-(3/2)] comes out similarly. --rwg
Then for negative m->m/(m-1): EllipticK[-16 EllipticEta[q^4]^8/EllipticEta[q]^8] -> Pi EllipticEta[q]^4/(2 EllipticEta[q^2]^2)
E.g., EllipticK[-((2 + GoldenRatio^(3/2))/(-2 + Sqrt[2 + Sqrt[5]]))] -> 5/21 5^(5/8) Sqrt[(2 (2 (-5 + Sqrt[5]) + Sqrt[10 (1 + Sqrt[5])]))/Pi] Gamma[9/20] Gamma[41/20]
I wonder if there are any valuations for less exotic "K-ands". There's a quadratic transformation leading to K(17-12 rt2) (Little League Home Plate! http://mathworld.wolfram.com/HomePlate.html) But I doubt it would simplify any of these weirdos. --rwg
On Wed, Dec 15, 2010 at 6:33 PM, Bill Gosper <billgosper@gmail.com> wrote:
For a horizontally swung pendulum (thetamax = pi/2), the Fourier series for am gives instead a fundamental of 4 sech(pi/2) = 1.594, exceeding pi/2 by < 2%. The "sechand" pi/2 fell out of some EllipticKs whose values were among the few currently known to FunctionExpand.
But my more-or-less systematic collection of special values of eta provides an abundance of K values, because
EllipticK[EllipticTheta[2, 0, q]^4/EllipticTheta[3, 0, q]^4] -> 1/2 Pi EllipticTheta[3, 0, q]^2 , i.e.,
EllipticK[(16 EllipticEta[q]^8 EllipticEta[q^4]^16)/EllipticEta[q^2]^24] -> Pi EllipticEta[q^2]^10/(2 EllipticEta[q]^4 EllipticEta[q^4]^4) ,
where EllipticEta[q_] -> EllipticTheta[1, \[Pi]/3, q^(1/6)]/Sqrt[3] ,
I.e., DedekindEta[tau]->EllipticEta[Exp[2*I*Pi*tau]]
E.g., EllipticK[1/2 + Sqrt[2] 3^(1/4) (-2 + Sqrt[3])] -> Gamma[1/4]^2/(2 3^(3/4) (-1 + Sqrt[3]) Sqrt[2 Pi]),
EllipticK[1/2 - Sqrt[2] 3^(1/4) (-2 + Sqrt[3])] -> 3^(1/4) Gamma[1/4]^2/(2 (-1 + Sqrt[3]) Sqrt[2 Pi]),
EllipticK[1 - (-2 + Sqrt[2] - Sqrt[3] + Sqrt[6])^4] -> 3^(3/4) Gamma[1/3]^3)/ (2 2^(5/6) (1 + Sqrt[3])^(5/2) Sqrt[-3 - 4 Sqrt[2] + 5 Sqrt[3]] Pi)
EllipticK[8 2^(1/4) (9 - 4 2^(1/4) - 3 Sqrt[2]))/(-1 + Sqrt[2])^6] -> (-1 + Sqrt[2])^(5/2) Gamma[1/4]^2/(4 Sqrt[2 (9 - 4 2^(1/4) - 3 Sqrt[2]) Pi])
EllipticK[-16 + 12 Sqrt[2]] -> 2 (2 + Sqrt[2]) Gamma[5/4]^2)/Sqrt[Pi] --rwg