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Re: request help

  • To: mathgroup at smc.vnet.net
  • Subject: [mg116202] Re: request help
  • From: "Sjoerd C. de Vries" <sjoerd.c.devries at gmail.com>
  • Date: Sat, 5 Feb 2011 05:43:54 -0500 (EST)
  • References: <iig76m$rl2$1@smc.vnet.net>

Well, I don't have an analytic approach. I just note that the plot of
the function seems smooth enough that it can be easily fitted, for
instance by the product of n and a polynome in r of order 4, as long
as you stay out of the forbidden areas of this plot:

ContourPlot[dis[r, n], {r, -1, 1}, {n, -1, 6}]

So, here some fit examples:

DynamicModule[{f, b, c},
 f[r_, n_] := n (b r^2 + c r^4);
 Manipulate[
  {b, c} = {b1, c1} /.
    FindFit[Table[{r, dis[r, n]}, {r, -0.2, 0.2, .01}],
     n b1 x^2 + n c1 x^4, {b1, c1}, x];
  Show[
   Plot[f[r, n], {r, -.2, .2}, PlotStyle -> Directive[Red, Dashed],
    PlotRange -> All],
   Plot[dis[r, n], {r, -.2, .2}, PlotRange -> All,
    PlotStyle -> Directive[Blue, DotDashed]]
   , ImageSize -> 500, Frame -> True],
  {n, 1, 5}, TrackedSymbols -> {n}
  ]
 ]

Given the fit, the road to the inverted function is now easy and left
as an exercise to the reader.

Cheers -- Sjoerd

On Feb 4, 7:43 am, Berihu Teklu <beri... at gmail.com> wrote:
> I need to invert a real function of two real variables Dis[r, n] with
> respect to the first variable r, while the second variable n is fixed.
> The function is rather difficult, that I couldn't invert it. this is
> kindly request you to write me any comments on the attached notebook.
>
> Many thanks for any help,
>
> Berihu
>
> Dis[r_, n_] :=
>  1/4 (2 (-2 + Sqrt[(1 + 2 n)^2]) Log[-2 + Sqrt[(1 + 2 n)^2]] -
>     2 (2 + Sqrt[(1 + 2 n)^2]) Log[
>       2 + Sqrt[(1 + 2 n)^2]] - (-2 +
>        Sqrt[(1 + 2 n)^2 Cosh[2 r]^2]) Log[-2 +
>        Sqrt[(1 + 2 n)^2 Cosh[2 r]^2]] + (2 +
>        Sqrt[(1 + 2 n)^2 Cosh[2 r]^2]) Log[
>       2 + Sqrt[(1 + 2 n)^2 Cosh[2 r]^2]] - (-2 +
>        Sqrt[((1 + 2 n)^2 (1 + 2 n -
>           2 Cosh[2 r])^2)/(-2 + (1 + 2 n) Cosh[2 r])^2]) Log[-2=
 +
>        Sqrt[((1 + 2 n)^2 (1 + 2 n -
>           2 Cosh[2 r])^2)/(-2 + (1 + 2 n) Cosh[2 r])^2]] + (2 +
>        Sqrt[((1 + 2 n)^2 (1 + 2 n -
>           2 Cosh[2 r])^2)/(-2 + (1 + 2 n) Cosh[2 r])^2]) Log[
>       2 + Sqrt[((1 + 2 n)^2 (1 + 2 n -
>           2 Cosh[2 r])^2)/(-2 + (1 + 2 n) Cosh[2 r])^2]])
>
> Solve[1/4 (2 (-2 + Sqrt[(1 + 2 n)^2]) Log[-2 + Sqrt[(1 + 2 n)^2]] -
>      2 (2 + Sqrt[(1 + 2 n)^2]) Log[
>        2 + Sqrt[(1 + 2 n)^2]] - (-2 +
>         Sqrt[(1 + 2 n)^2 Cosh[2 r]^2]) Log[-2 +
>         Sqrt[(1 + 2 n)^2 Cosh[2 r]^2]] + (2 +
>         Sqrt[(1 + 2 n)^2 Cosh[2 r]^2]) Log[
>        2 + Sqrt[(1 + 2 n)^2 Cosh[2 r]^2]] - (-2 +
>         Sqrt[((1 + 2 n)^2 (1 + 2 n -
>            2 Cosh[2 r])^2)/(-2 + (1 + 2 n) Cosh[2 r])^2]) Log=
[-2 +
>         Sqrt[((1 + 2 n)^2 (1 + 2 n -
>            2 Cosh[2 r])^2)/(-2 + (1 + 2 n) Cosh[2 r])^2]] + (=
2 +
>         Sqrt[((1 + 2 n)^2 (1 + 2 n -
>            2 Cosh[2 r])^2)/(-2 + (1 + 2 n) Cosh[2 r])^2]) Log=
[
>        2 + Sqrt[((1 + 2 n)^2 (1 + 2 n -
>            2 Cosh[2 r])^2)/(-2 + (1 + 2 n) Cosh[2 r])^2]]) =
==
>   Dis[r, n], r]



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