Re: Simplify[ {Re[Sqrt[-1 + eta^2]], Im[Sqrt[-1 + eta^2]]}, eta<1]
- To: mathgroup at smc.vnet.net
- Subject: [mg50628] Re: [mg50617] Simplify[ {Re[Sqrt[-1 + eta^2]], Im[Sqrt[-1 + eta^2]]}, eta<1]
- From: Andrzej Kozlowski <andrzej at akikoz.net>
- Date: Wed, 15 Sep 2004 01:49:20 -0400 (EDT)
- References: <200409130619.CAA14342@smc.vnet.net> <8C2E6168-0558-11D9-A0AA-000A95B4967A@akikoz.net>
- Sender: owner-wri-mathgroup at wolfram.com
On 13 Sep 2004, at 16:43, Andrzej Kozlowski wrote:
> On 13 Sep 2004, at 15:19, peteraptaker wrote:
>
>> *This message was transferred with a trial version of CommuniGate(tm)
>> Pro*
>> Have I missed something - my apologies if this is answered in a FAQ
>> I want to make the simple Re and Im parts simplify properly?
>>
>> test =
>> {Re[Sqrt[-1 + eta^2]], Im[Sqrt[-1 + eta^2]]}
>>
>> FullSimplify[test, eta > 1]
>> gives*{Sqrt[-1 + eta^2], 0}
>>
>> But
>> FullSimplify[test, eta < 1]
>> gives
>> {Re[Sqrt[-1 + eta^2]], Im[Sqrt[-1 + eta^2]]}
>>
>> Needs["Algebra`ReIm`"] does not seem to help
>>
>> Real numbers demonstrate what should happen:
>> test) /. {{eta -> 0.1}, {eta -> 2}}
>> {{0., 0.99498743710662}, {Sqrt[3], 0}}
>>
>>
>
> There is nothing really strange here, Mathematica simply can't give a
> single simple expression that would cover all the cases that arise. So
> you have to split it yourself, for example:
>
>
> FullSimplify[test, eta < -1]
>
>
> {Sqrt[eta^2 - 1], 0}
>
> FullSimplify[test, eta == -1]
>
> {0, 0}
>
>
> FullSimplify[test, -1 < eta < 1]
>
> {0, Sqrt[1 - eta^2]}
>
>
> FullSimplify[test, eta == 1]
>
>
> {0, 0}
>
>
> FullSimplify[test, 1 <= eta]
>
>
> {Sqrt[eta^2 - 1], 0}
>
>
> or, you can combine everything into just two cases:
>
> FullSimplify[test, eta $B":(B Reals && Abs[eta] < 1]
>
> {Re[Sqrt[eta^2 - 1]], Im[Sqrt[eta^2 - 1]]}
>
>
> FullSimplify[test, eta $B":(B Reals && Abs[eta] >= 1]
>
> {Sqrt[eta^2 - 1], 0}
>
> In fact you do not really need FullSimplify, simple Simplify will do
> just as well.
>
I now noticed that I had somehow pasted the wrong cells at the end of
the message above; the two cases actually give:
FullSimplify[test, eta $B":(B Reals && Abs[eta] < 1]
{0, Sqrt[1 - eta^2]}
FullSimplify[test, eta $B":(B Reals && Abs[eta] >= 1]
{Sqrt[eta^2 - 1], 0}
Andrzej
- References:
- Simplify[ {Re[Sqrt[-1 + eta^2]], Im[Sqrt[-1 + eta^2]]}, eta<1]
- From: psa@laplacian.co.uk (peteraptaker)
- Simplify[ {Re[Sqrt[-1 + eta^2]], Im[Sqrt[-1 + eta^2]]}, eta<1]