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