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