Re: How to simplify to a result that is real

*To*: mathgroup at smc.vnet.net*Subject*: [mg50810] Re: How to simplify to a result that is real*From*: Paul Abbott <paul at physics.uwa.edu.au>*Date*: Wed, 22 Sep 2004 00:11:19 -0400 (EDT)*Organization*: The University of Western Australia*References*: <cidt38$brv$1@smc.vnet.net> <200409180948.FAA00572@smc.vnet.net> <cion2b$r9a$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

In article <cion2b$r9a$1 at smc.vnet.net>, Richard Chen <richard at doubleprime.com> wrote: > This time, mathematica comes back with an expression involving > ArcTan[u,v] which is just ArcTan[v/u], by direct inspection. > So the 2 terms involving different ArcTan are actually the same. > But I cannot easily make Mathematica to recognize that they > are the same. I'm not sure if I follow you here, but ArcTan[u,v] is equal to ArcTan[v/u] only if u > 0. For example, try FullSimplify[ArcTan[u,v] == ArcTan[v/u], u > 0] However, if u < 0 then ArcTan[u,v] - ArcTan[v/u] = Sign[v] Pi (though I cannot see how to get FullSimplify to deduce this fact). Cheers, Paul > Even if I use the rule ArcTan[u_,v_]->ArcTan[v/u] > it still does not think the 2 expressions are the same. It is easier > to simply copy and paste an anwser than manipulate mathematica > to get a simpler result. > > Perhaps future versions of mathematica will be smarter and does not > require as much intervention from the user. > > Thanks > > Richard > > On Sat, Sep 18, 2004 at 05:48:55AM -0400, Peter Valko wrote: > > Richard Chen <richard at doubleprime.com> wrote in message > > news:<cidt38$brv$1 at smc.vnet.net>... > > > The command: > > > > > > Integrate[1/(1 + e Cos[t]), {t, 0, a}, > > > Assumptions -> {-1 < e < 1, 0 < a < Pi}] > > > > > > leads to a complex valued result. I could not make > > > mathematica to render the result in a form that is > > > purely real. ComplexExpand, Refine all do not seem to work. > > > > > > Does anyone know how to make mathematica to simplify this > > > result into a real form? > > > > > > Thanks for any info. > > > > > > Richard > > > > > > > > Richard, > > > > I think this will work: > > > > > > ch = Integrate[1/(1 + e Cos[t]), {t, 0, a}, Assumptions -> {-1 < e < > > 1, 0 < a < Pi}] > > > > FullSimplify[ComplexExpand[ch, TargetFunctions -> {Re, Im}], {-1 < e < > > 1, 0 < a < Pi}] > > > > > > The result is > > > > (-2*ArcTan[((-1 + e)*Tan[a/2])/Sqrt[1 - e^2]])/Sqrt[1 - e^2] > > > > > > Peter > > > -- Paul Abbott Phone: +61 8 6488 2734 School of Physics, M013 Fax: +61 8 6488 1014 The University of Western Australia (CRICOS Provider No 00126G) 35 Stirling Highway Crawley WA 6009 mailto:paul at physics.uwa.edu.au AUSTRALIA http://physics.uwa.edu.au/~paul

**References**:**Re: How to simplify to a result that is real***From:*p-valko@tamu.edu (Peter Valko)