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Re: Exploiting relationships in manipulations: example with conjugate relationship
*To*: undisclosed-recipients:;
*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
*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com
*Delivered-to*: l-mathgroup@wolfram.com
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*Newsgroups*: comp.soft-sys.math.mathematica
*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[2][-1] + ff[2][1])
to become
-2 Sqrt[(2 =CF=80)/15] Im(ff[2][1]).
Sune
On 28 Jun, 2013, at 17:42 , Bob Hanlon <hanlonr357 at gmail.com> wrote:
> tel = {2/15 Sqrt[
> =CF=80] (5 ff[0][0] + 2 Sqrt[5] ff[2][0]), -I Sqrt[(2 =CF=80)/15] (ff[2][-1] +
> ff[2][1]),
> 1/15 Sqrt[
> =CF=80] (10 ff[0][0] -
> Sqrt[5] (Sqrt[6] ff[2][-2] + 2 ff[2][0] + Sqrt[6] ff[2][2])),
> Sqrt[(2 =CF=80)/15] (ff[2][-1] - ff[2][1]), -I Sqrt[(2 =CF=80)/15] (ff[2][-2] -
> ff[2][2]),
> 1/15 Sqrt[
> =CF=80] (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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