Re: get rid of I in the result of an integral
- To: mathgroup at smc.vnet.net
- Subject: [mg116525] Re: get rid of I in the result of an integral
- From: Bob Hanlon <hanlonr at cox.net>
- Date: Fri, 18 Feb 2011 04:35:24 -0500 (EST)
Tell it to avoid ArcSec and ArcCsc
Assuming[0 < x < 1,
FullSimplify[
Integrate[1/((1 + y)^2 Sqrt[-x^2 + y^2]), {y, x, Infinity}],
ComplexityFunction -> (LeafCount[#] +
100 Count[#, _ArcSec | _ArcCsc, {0, Infinity}] &)]]
(-1 + x^2 + Sqrt[1 - x^2]*ArcSech[x])/(-1 + x^2)^2
Bob Hanlon
---- "Ruth Lazkoz S=C3=A1ez" <ruth.lazkoz at ehu.es> wrote:
==========================
Thanks for the tip, I would however like to get the result expressed
using ArcSech[x] without having to tell Mathematica explictly to do
/.ArcSec[x]->-I ArcSech[x] at the end
El 17/02/11 14:04, Bob Hanlon escribi=C3=B3:
> sol1 = Assuming[{Element[x, Reals], 0< x< 1},
> Integrate[1/((1 + y)^2 Sqrt[-x^2 + y^2]), {y, x, Infinity}]]
>
> (-1 + x^2 - I*Sqrt[1 - x^2]*ArcSec[x])/(-1 + x^2)^2
>
> sol2 = Assuming[{Element[x, Reals], 0< x< 1},
> FullSimplify[
> Integrate[1/((1 + y)^2 Sqrt[-x^2 + y^2]), {y, x, Infinity}] //
> TrigToExp]]
>
> (-1 + x^2 + Sqrt[1 - x^2]*Log[(1 + Sqrt[1 - x^2])/x])/(-1 + x^2)^2
>
> Using the form Assuming[{ _ }, _ ], makes the assumtions available to both FullSimplify and Integrate without having to repeat the assumptions.
>
> Assuming[{Element[x, Reals], 0< x< 1},
> FullSimplify[sol1 - sol2 == 0]]
>
> True
>
>
> Bob Hanlon
>
> ---- "Ruth Lazkoz S=C3=A1ez"<ruth.lazkoz at ehu.es> wrote:
>
> ==========================
> Hi,
>
> When I do Integrate[1/((1 + y)^2 Sqrt[-x^2 + y^2]), {y, x, Infinity},
> Assumptions -> x^2< 1&& 1> x> 0&& Element[x, Reals]]
>
> I get
>
> (-1 + x^2 - I Sqrt[1 - x^2] ArcSec[x])/(-1 + x^2)^2
>
> and I would like the result expresed in terms of ArcSech[x] so that I
> does not appear.
>
> I know I can tell Mathematica to do the replacement afterwards, but I
> want to show the result to a colleague who is not so familiar with
> mathematica, and if it were possible to get the result in one go just by
> adding some extra assumption or so, it would be more convincing.
>
> Help will be appreciated. Best,
>
> Ruth