Re: How to get a positive solution from Solve Command
- To: mathgroup at smc.vnet.net
- Subject: [mg67825] Re: How to get a positive solution from Solve Command
- From: "David W. Cantrell" <DWCantrell at sigmaxi.net>
- Date: Sun, 9 Jul 2006 04:50:52 -0400 (EDT)
- References: <e8nte8$k9j$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
abdou.oumaima at hotmail.com wrote:
> By now, I fix the problem as that:
> f[ξ_] := Normal[Series[1 - c ArcTanh[ξ]/ξ, {ξ, 0,
> 2}]] Solve[f[ξ] == 0, ξ];
> ξ = ξ /. % [[2]];
>
> What do you think?
Given some c in the interval [0,1], you're wanting to solve
1 - c ArcTanh[x]/x == 0 for x.
(Is there some particular application in which that equation arises?)
Unless c happens to be close to 1, the basic method you proposed will give
highly inaccurate answers.
Your equation is equivalent to ArcTanh[x]/x == 1/c, and so if you could
somehow invert the function ArcTanh[x]/x, you could get a neat solution.
As you probably realized, that function cannot be inverted precisely in
closed form. Nonetheless, its inverse can be approximated simply by
Sqrt[1 - Exp[(1 - y)(2 + 1/y)]]
Replacing y with 1/c then gives an approximate solution of your equation
for x:
Sqrt[1 - Exp[(1 - 1/c)(2 + c)]]
For that approximation, |error| < 0.0009 .
David W. Cantrell