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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]]



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