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Re: Incongruence? hmm...

  • To: mathgroup at smc.vnet.net
  • Subject: [mg102760] Re: Incongruence? hmm...
  • From: garrido at ruth.upc.edu
  • Date: Wed, 26 Aug 2009 07:44:17 -0400 (EDT)

Hi Filippo,

FourierCosCoefficient[
 1/720 (8 \[Pi]^4 - 60 \[Pi]^2 x^2 + 60 \[Pi] Abs[x]^3 - 15 x^4), x, m]

1/m^4


1/720 (8 \[Pi]^4 - 60 \[Pi]^2 x^2 + 60 \[Pi] Abs[x]^3 - 15 x^4) /.
 x -> Mod[x, 2 Pi]

1/720 (8 \[Pi]^4 + 60 \[Pi] Abs[Mod[x, 2 \[Pi]]]^3 -
   60 \[Pi]^2 Mod[x, 2 \[Pi]]^2 - 15 Mod[x, 2 \[Pi]]^4)

Plot[{Sum[Cos[m x]/m^4, {m, 1, \[Infinity]}],
   1/720 (8 \[Pi]^4 + 60 \[Pi] Abs[Mod[x, 2 \[Pi]]]^3 -
      60 \[Pi]^2 Mod[x, 2 \[Pi]]^2 - 15 Mod[x, 2 \[Pi]]^4)}, {x, -10,
   10}];


Therefore,

Sum[Cos[m x]/m^4, {m, 1, \[Infinity]}] =
 1/720 (8 \[Pi]^4 + 60 \[Pi] Abs[Mod[x, 2 \[Pi]]]^3 -
    60 \[Pi]^2 Mod[x, 2 \[Pi]]^2 -
    15 Mod[x, 2 \[Pi]]^4) for everything  x Real.


   Regards,
J.L.Garrido


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