Re: Exploiting relationships in manipulations: example

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• Subject: [mg131342] Re: Exploiting relationships in manipulations: example
• From: Bob Hanlon <hanlonr357 at gmail.com>
• Date: Sat, 29 Jun 2013 04:58:31 -0400 (EDT)
• Approved: Steven M. Christensen <steve@smc.vnet.net>, Moderator
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• References: <20130628081233.8C1D769C6@smc.vnet.net>

```tel = {2/15 Sqrt[=F0] (5 ff[0][0] + 2 Sqrt[5] ff[2][0]),
-I Sqrt[(2 =F0)/15] (ff[2][-1] + ff[2][1]),
1/15 Sqrt[=F0] (10 ff[0][0] - Sqrt[5]*
(Sqrt[6] ff[2][-2] + 2 ff[2][0] + Sqrt[6] ff[2][2])),
Sqrt[(2 =F0)/15] (ff[2][-1] - ff[2][1]),
-I Sqrt[(2 =F0)/15] (ff[2][-2] - ff[2][2]),
1/15 Sqrt[=F0] (10 ff[0][0] + Sqrt[5]*
(Sqrt[6] ff[2][-2] - 2 ff[2][0] + Sqrt[6] ff[2][2]))};

FullSimplify[tel /. Cases[tel,
ff[l_][m_?Negative] :>
(ff[l][m] :> Conjugate[ff[l][-m]]*(-1)^m),
Infinity] // Union] /. (Re[x_] - x_) :> -Im[x]

{(2/15)*Sqrt[Pi]*(5*ff[0][0] +
2*Sqrt[5]*ff[2][0]),
-2*Sqrt[(2*Pi)/15]*Re[ff[2][1]],
-2*I*Sqrt[(2*Pi)/15]*Im[ff[2][1]],
2*I*Sqrt[(2*Pi)/15]*Im[ff[2][2]],
(2/15)*Sqrt[Pi]*
(Sqrt[30]*Re[ff[2][2]] +
5*ff[0][0] - Sqrt[5]*ff[2][0]),
(-(2/15))*Sqrt[Pi]*
(Sqrt[30]*Re[ff[2][2]] -
5*ff[0][0] + Sqrt[5]*ff[2][0])}

Bob Hanlon

On Fri, Jun 28, 2013 at 1:30 PM, Sune Jespersen <sunenj at gmail.com> wrote:

> 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 =F0)/15] (ff[2][-1] + ff[2][1])
> to become
> -2 Sqrt[(2 =F0)/15] Im(ff[2][1]).
>
> Sune
>
> On 28 Jun, 2013, at 17:42 , Bob Hanlon <hanlonr357 at gmail.com> wrote:
>
> tel = {2/15 Sqrt[
>      =F0] (5 ff[0][0] + 2 Sqrt[5] ff[2][0]), -I Sqrt[(2 =F0)/15] (ff[2][-1] +
>       ff[2][1]),
>    1/15 Sqrt[
>      =F0] (10 ff[0][0] -
>       Sqrt[5] (Sqrt[6] ff[2][-2] + 2 ff[2][0] + Sqrt[6] ff[2][2])),
>    Sqrt[(2 =F0)/15] (ff[2][-1] - ff[2][1]), -I Sqrt[(2 =F0)/15] (ff[2][-2] -
>       ff[2][2]),
>    1/15 Sqrt[
>      =F0] (10 ff[0][0] +
>       Sqrt[5] (Sqrt[6] ff[2][-2] - 2 ff[2][0] + Sqrt[6] ff[2][2]))};
>
> 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[0][0] +
>         2*Sqrt[5]*ff[2][0]), 0, (2/15)*Sqrt[Pi]*
>      (5*ff[0][0] - Sqrt[5]*(ff[2][0] +
>              Sqrt[6]*ff[2][2])), -2*Sqrt[(2*Pi)/15]*
>      ff[2][1], 0, (2/15)*Sqrt[Pi]*
>      (5*ff[0][0] - Sqrt[5]*ff[2][0] +
>         Sqrt[30]*ff[2][2])}
>
> 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[0][0] +
>         2*Sqrt[5]*ff[2][0]), (-I)*Sqrt[(2*Pi)/15]*
>      (ff[2][-1] + ff[2][1]), (1/15)*Sqrt[Pi]*
>      (10*ff[0][0] - Sqrt[5]*(Sqrt[6]*ff[2][-2] +
>              2*ff[2][0] + Sqrt[6]*ff[2][2])),
>    Sqrt[(2*Pi)/15]*(ff[2][-1] - ff[2][1]),
>    (-I)*Sqrt[(2*Pi)/15]*(ff[2][-2] - ff[2][2]),
>    (1/15)*Sqrt[Pi]*(10*ff[0][0] +
>         Sqrt[5]*(Sqrt[6]*ff[2][-2] - 2*ff[2][0] +
>              Sqrt[6]*ff[2][2]))}
>
> % === 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[0][0]+2 Sqrt[5] ff[2][0]),-I Sqrt[(2
>> \[Pi])/15] (ff[2][-1]+ff[2][1]),1/15 Sqrt[\[Pi]] (10 ff[0][0]-Sqrt[5]
>> (Sqrt[6] ff[2][-2]+2 ff[2][0]+Sqrt[6] ff[2][2])),Sqrt[(2 \[Pi])/15]
>> (ff[2][-1]-ff[2][1]),-I Sqrt[(2 \[Pi])/15] (ff[2][-2]-ff[2][2]),1/15
>> Sqrt[\[Pi]] (10 ff[0][0]+Sqrt[5] (Sqrt[6] ff[2][-2]-2 ff[2][0]+Sqrt[6]
>> ff[2][2]))}
>>
>> 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[2][-1]+ff[2][1])
>>
>> to
>>
>> -Sqrt[(2 \[Pi])/15] 2*Im(ff[2][1])
>>
>>
>> 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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