Re: Elliptic integral problems in mma

*To*: mathgroup at yoda.physics.unc.edu*Subject*: Re: Elliptic integral problems in mma*From*: keiper*Date*: Wed, 28 Oct 92 10:43:48 CST

> Note that the parameter m > 1 (periodic solution). > The case m < 1 corresponds to a strictly increasing > y=y[t] (initial push z0 large enough), and in this > case there is only one curve There are indeed two cases and the code defining JacobiAmplitude[ ] (in StartUp`Elliptic`) is wrong. It tries to deal with the more complicated case of m < 1 (which it gets right), but messes up on the much easier case of m > 1. Below is the fix which should be put in the file StartUp`Elliptic` near line 1008. (Delete the old code.) I take full responsibility for this error and appologize for any inconvenience it may have caused. Jerry B. Keiper keiper at wri.com (* ----------------------- JacobiAmplitude ------------------------ *) JacobiAmplitude[u_, 0] := u; JacobiAmplitude[u_, 1] := 2 ArcTan[Exp[u]] - Pi/2; JacobiAmplitude[u_?NumberQ, m_?NumberQ] := Module[{am = ArcSin[JacobiSN[u,m]], pi, s, k2, kp, km}, k2 = 2 $EllipK[N[m, Precision[am]]]; s = EllipticF[am, m]; pi = N[Pi, Accuracy[am]+2]; km = Re[(u - s)/k2]; kp = Re[(u + s)/k2]; If[Abs[km - Round[km]] > Abs[kp - Round[kp]], Round[kp] pi - am, (* else *) Round[km] pi + am] ] /; (Precision[{u, m}] < Infinity && Head[m] =!= Complex && Abs[m] < 1) JacobiAmplitude[u_?NumberQ, m_?NumberQ] := ArcSin[JacobiSN[u,m]] /; Precision[{u, m}] < Infinity JacobiAmplitude/: Derivative[1,0][JacobiAmplitude] = JacobiDN InverseFunction[JacobiAmplitude, 1, 2] = EllipticF InverseFunction[EllipticF, 1, 2] = JacobiAmplitude