Re: problem with the usage or implemetation of EllipticF[x,m] and

*To*: mathgroup at smc.vnet.net*Subject*: [mg110359] Re: problem with the usage or implemetation of EllipticF[x,m] and*From*: Luc Roy <luc.rg.roy at gmail.com>*Date*: Tue, 15 Jun 2010 02:29:19 -0400 (EDT)*References*: <hv3nhj$r3$1@smc.vnet.net>

On Jun 13, 6:53 pm, Luc Roy <luc.rg.... at gmail.com> wrote: > FullSimplify[JacobiAmplitude [Series[EllipticF[x, m], {x, 0, 10}], m]] > > should give x + O[x]^10 because the two functions are the inverse of > one another. > > however it gives > > x - (m x^3)/3 + 1/30 m (2 + 3 m) x^5 - 1/630 m (4 + m (39 + 16 m)) x^7 > + ( m (8 + 3 m (120 + 13 m (20 + 3 m))) x^9)/22680 > > Am I missing something > or > Is there an implementation problem with Mathematica > > Note : the usage of series is very important. > I know the following works and that is not what I need to resolve or > understand. > PowerExpand[JacobiAmplitude [EllipticF[x, m], m]] > Answer: x I found the problem. The following evaluates to the series decribed above. FullSimplify[InverseSeries[EllipticF [Series[EllipticF[x, m], {x, 0, 10}], m]]] This implies that JacobiAmplitude[[fx]] is evaluated as InverseSeries[EllipticF[f[x], m]] where f[x] is evaluated first, EllipticF second and InverseSeries third. However, the InverseSeries only applies to EllipticF and not to f[x] Mathematica has a bad implementation for JacobiAmplitude. Instead of JacobiAmplitude use ArcSin[JacobiSN[x,m]]