Re: How to simplify to a result that is real

• To: mathgroup at smc.vnet.net
• Subject: [mg50735] Re: How to simplify to a result that is real
• From: p-valko at tamu.edu (Peter Valko)
• Date: Sat, 18 Sep 2004 05:48:55 -0400 (EDT)
• References: <cidt38\$brv\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```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

```

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