On Thu, Dec 2, 2010 at 4:26 AM, Bill Gosper <billgosper@gmail.com> wrote:

(Retreating to the relative safety of mathematics.) In the special case of "half overswing", i.e., horizontal extrema, the Fourier series comes out particularly simple (though not particularly simply): theta[t]== 2*JacobiAmplitude[(Sqrt[g/L]*t)/Sqrt[2], 2] == 4*Sum[(Sech[(1/2)*(-1 + 2*k)*Pi]* Sin[((-1 + 2*k)*Sqrt[g/L]*t*Gamma[-(1/4)]^2)/(16*Sqrt[Pi])])/(-1 + 2*k), {k, Infinity}]

g Sqrt[-] t L 2 JacobiAmplitude[---------, 2] == Sqrt[2]

g 1 2 (-1 + 2 k) Sqrt[-] t Gamma[-(-)] 1 L 4 Sech[- (-1 + 2 k) Pi] Sin[---------------------------------] 2 16 Sqrt[Pi] 4 Sum[------------------------------------------------------------, -1 + 2 k {k, Infinity}]

Does anyone recognize this identity? The Fourier series for general swing amplitude looks a bit tougher.

Poorly tested and fishy looking, but \[Theta][t] == 2*JacobiAmplitude[Sqrt[g/L]*t*Sin[Subscript[\[Theta], max]/2], Csc[Subscript[\[Theta], max]/2]^2] == 2*Sum[(Sech[((-(1/2) + n)*Pi* EllipticK[1 - Csc[Subscript[\[Theta], max]/2]^2])/ Re[EllipticK[Csc[Subscript[\[Theta], max]/2]^2]]]* Sin[(Sqrt[g/L]*(-(1/2) + n)*Pi*t* Sin[Subscript[\[Theta], max]/2])/ Re[EllipticK[Csc[Subscript[\[Theta], max]/2]^2]]])/ (-(1/2) + n), {n, 1, Infinity}] seems to work. I.e., JacobiAmplitude[u, m] == Sum[(Sech[((-(1/2) + n)*Pi*EllipticK[1 - m])/Re[EllipticK[m]]]* Sin[((-(1/2) + n)*Pi*u)/Re[EllipticK[m]]])/ (-(1/2) + n), {n, 1, Infinity}] This is suggested by the usual (Lambert) am series JacobiAmplitude[u, m] == (Pi*u)/(2*EllipticK[m]) + 2*Sum[(q^n*Sin[(n*Pi*u)/EllipticK[m]])/(n*(1 + q^(2*n))), {n, Infinity}] with q == Exp[-\[Pi]*EllipticK[1 - m]/EllipticK[m]] which *looks* like a Fourier series except a) there's a linear term b) q is complex in the oscillatory (underswung) case, concealing additional triggery. c) it only converges over +- a quarter period. If the former holds up, then we need to know why, and why it isn't widelier known. --rwg (Overseas readers: Do not learn English from this.)

--rwg