Re: How to simplify to a result that is real

*To*: mathgroup at smc.vnet.net*Subject*: [mg50742] Re: [mg50679] How to simplify to a result that is real*From*: DrBob <drbob at bigfoot.com>*Date*: Sat, 18 Sep 2004 05:49:17 -0400 (EDT)*References*: <200409170515.BAA11699@smc.vnet.net>*Reply-to*: drbob at bigfoot.com*Sender*: owner-wri-mathgroup at wolfram.com

This may not help you, but here's a reduction of the result to a clearly real-valued expression. First, look at this expansion of the Log of a Complex number: ComplexExpand[Log[x + I*y]] First[%] I*Arg[x + I*y] + (1/2)*Log[x^2 + y^2] I*Arg[x + I*y] Later I'll apply this to a complex number whose Norm is one, so that Log[x^2 + y^2]==0. Here's the Integrate output: one = Integrate[1/(1 + e*Cos[t]), {t, 0, a}, Assumptions -> {-1 < e < 1, 0 < a < Pi}] -((1/Sqrt[1 - e^2])*(I*Log[Sqrt[1 - e^2] - I*(-1 + e)*Tan[ a/2]])) + (1/Sqrt[1 - e^2])*(I*Log[Sqrt[1 - e^2] + I*(-1 + e)*Tan[a/2]]) Next I factor Sqrt[1 - e^2]/I out of the expression. Later I'll undo this. two = (#1*(Sqrt[1 - e^2]/I) & ) /@ one -Log[Sqrt[1 - e^2] - I*(-1 + e)*Tan[a/2]] + Log[Sqrt[1 - e^2] + I*(-1 + e)*Tan[a/2]] Transform the difference of Logs as follows: three = two /. -Log[x_] + Log[y_] -> Log[x/y] Log[(Sqrt[1 - e^2] - I*(-1 + e)*Tan[a/2])/ (Sqrt[1 - e^2] + I*(-1 + e)*Tan[a/2])] The argument of Log is a number divided by its Conjugate, so its Norm is one, hence applying the preliminary result and multiplying by I (undoing part of the first factorization above) gives: four = three /. Log[x_] ->I*I*Arg[x] -Arg[(Sqrt[1 - e^2] - I*(-1 + e)*Tan[a/2])/(Sqrt[1 - e^2] + I*(-1 + e)*Tan[a/2])] Divide by Sqrt[1 - e^2] to undo the factorization and we have: four/Sqrt[1 - e^2] Arg[(Sqrt[1 - e^2] - I*(-1 + e)*Tan[a/2])/(Sqrt[1 - e^2] + I*(-1 + e)*Tan[a/2])]/Sqrt[1 - e^2] That expression involves I, it's true, but Arg is real-valued, so the value is certainly Real. Bobby On Fri, 17 Sep 2004 01:15:42 -0400 (EDT), Richard Chen <richard at doubleprime.com> wrote: > 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 > > > -- DrBob at bigfoot.com www.eclecticdreams.net

**References**:**How to simplify to a result that is real***From:*Richard Chen <richard@doubleprime.com>