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Exploiting relationships in manipulations: example with conjugate relationship

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  • Subject: [mg131319] Exploiting relationships in manipulations: example with conjugate relationship
  • From: Sune <sunenj at gmail.com>
  • Date: Fri, 28 Jun 2013 04:12:33 -0400 (EDT)
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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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