       Re: Exploiting relationships in manipulations: example with conjugate relationship

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• Subject: [mg131340] Re: Exploiting relationships in manipulations: example with conjugate relationship
• From: Sune Jespersen <sunenj at gmail.com>
• Date: Sat, 29 Jun 2013 04:57:51 -0400 (EDT)
• Approved: Steven M. Christensen <steve@smc.vnet.net>, Moderator
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• References: <20130628081233.8C1D769C6@smc.vnet.net> <CAEtRDSezcQXsrmSzzWO1yp-CPjGSeNncd9cfF6soHb3GYQvXBQ@mail.gmail.com>

```Thanks. I meant  ff[l][m] == Conjugate[ff[l][-m]]*(-1)^m). It seems your solution in this case produces an output fully identical (unchanged) to tel.
For example, I wanted the 2nd element of tel
I Sqrt[(2 =CF=80)/15] (ff[-1] + ff)
to become
-2 Sqrt[(2 =CF=80)/15] Im(ff).

Sune

On 28 Jun, 2013, at 17:42 , Bob Hanlon <hanlonr357 at gmail.com> wrote:

> tel = {2/15 Sqrt[
>      =CF=80] (5 ff + 2 Sqrt ff), -I Sqrt[(2 =CF=80)/15] (ff[-1] +
>       ff),
>    1/15 Sqrt[
>      =CF=80] (10 ff -
>       Sqrt (Sqrt ff[-2] + 2 ff + Sqrt ff)),
>    Sqrt[(2 =CF=80)/15] (ff[-1] - ff), -I Sqrt[(2 =CF=80)/15] (ff[-2] -
>       ff),
>    1/15 Sqrt[
>      =CF=80] (10 ff +
>       Sqrt (Sqrt ff[-2] - 2 ff + Sqrt ff))};
>
> In your text you state ff[l][m] == ff[l][-m] (-1)^m
>
> Simplify[tel,
>  Union[Cases[tel, ff[_][_], Infinity]] /.
>   ff[l_][m_] -> (ff[l][m] == ff[l][-m] (-1)^m)]
>
> {(2/15)*Sqrt[Pi]*(5*ff +
>         2*Sqrt*ff), 0, (2/15)*Sqrt[Pi]*
>      (5*ff - Sqrt*(ff +
>              Sqrt*ff)), -2*Sqrt[(2*Pi)/15]*
>      ff, 0, (2/15)*Sqrt[Pi]*
>      (5*ff - Sqrt*ff +
>         Sqrt*ff)}
>
> However, in your code you use ff[l][m] == =
Conjugate[ff[l][-m]]*(-1)^m)
>
> Simplify[tel,
>  Union[Cases[tel, ff[_][_], Infinity]] /.
>   ff[l_][m_] -> (ff[l][m] == Conjugate[ff[l][-m]]*(-1)^m)]
>
> {(2/15)*Sqrt[Pi]*(5*ff +
>         2*Sqrt*ff), (-I)*Sqrt[(2*Pi)/15]*
>      (ff[-1] + ff), (1/15)*Sqrt[Pi]*
>      (10*ff - Sqrt*(Sqrt*ff[-2] +
>              2*ff + Sqrt*ff)),
>    Sqrt[(2*Pi)/15]*(ff[-1] - ff),
>    (-I)*Sqrt[(2*Pi)/15]*(ff[-2] - ff),
>    (1/15)*Sqrt[Pi]*(10*ff +
>         Sqrt*(Sqrt*ff[-2] - 2*ff +
>              Sqrt*ff))}
>
> % === tel
>
> True
>
>
> Bob Hanlon
>
>
>
> On Fri, Jun 28, 2013 at 4:12 AM, Sune <sunenj at gmail.com> wrote:
> Hey all.
>
> I'm trying to get Mathematica to simplify a list of expressions involving complex symbolic variables with certain relations among them, and to take advantage of these relations while simplifying.
>
> To be more concrete, I could have a list such as
>
> tel={2/15 Sqrt[\[Pi]] (5 ff+2 Sqrt ff),-I Sqrt[(2 \[Pi])/15] (ff[-1]+ff),1/15 Sqrt[\[Pi]] (10 ff-Sqrt (Sqrt ff[-2]+2 ff+Sqrt ff)),Sqrt[(2 \[Pi])/15] (ff[-1]-ff),-I Sqrt[(2 \[Pi])/15] (ff[-2]-ff),1/15 Sqrt[\[Pi]] (10 ff+Sqrt (Sqrt ff[-2]-2 ff+Sqrt ff))}
>
> However, there's a conjugate symmetry among the variables ff[l][m] that would enable a simpler looking expression. Specifically, ff[l][m]=ff[l][-m] (-1)^m, and I would like to have Mathematica take advantage of that and reduce expressions such as
>
> I Sqrt[(2 \[Pi])/15] (ff[-1]+ff)
>
> to
>
> -Sqrt[(2 \[Pi])/15] 2*Im(ff)
>
>
> I've tried various combinations of ComplexExpand and FullSimplify;
>
> ComplexExpand[
>  FullSimplify[tel,
>   And @@ Flatten[
>     Table[ff[l][m] == Conjugate[ff[l][-m]]*(-1)^m, {l, 0, 4, 2}, {m,
>       0, l}]]], Flatten[Table[ff[l][m] , {l, 2, 4, 2}, {m, -l, l}]]]
>
> (And also version with the two outermost commands interchanged)
>  but it doesn't do what I want. Of course, it may be that the rules for simplify are such that my sought expression is not considered a simpler version of the same expression. Could that be the case? Otherwise, I'd appreciate any suggestions on how to implement relations such as these in manipulation of expressions.
>
> Thanks,
> Sune
>
>

```

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