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Re: failure to Integrate in orthogonal polynomials

  • To: mathgroup at smc.vnet.net
  • Subject: [mg67918] Re: failure to Integrate in orthogonal polynomials
  • From: Roger Bagula <rlbagula at sbcglobal.net>
  • Date: Fri, 14 Jul 2006 02:11:09 -0400 (EDT)
  • References: <e8nt4n$k7c$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Based on my experience that a unit sphere in an so(3) matrix gives a 
square that gives a projective plane I did
in the off diagonal elements:
M = {{0, LegendreP[n, x], -LegendreP[n + 1, x]},
    {-LegendreP[n, x], 0, LegendreP[n + 2, x]},
    {LegendreP[n + 1, x], -LegendreP[n + 2, x], 0}}
M.M
Unnormalized the functions  tend to grow larger as they are projected.
When presented in this way the result won't integrate:
norm[n_]=Sqrt[1/Integrate[LegendreP[n,x]^2,{x,-1,1}]]
g=Table[ParametricPlot3D[{norm[n+1]*norm[n+2]*LegendreP[n+1,x]*LegendreP[n+2,\
x],norm[n]*norm[n+2]*LegendreP[n,x]*LegendreP[n+2,x],norm[n]*norm[n+1]*\
LegendreP[n,x]*LegendreP[n+1,x]},{x,-1,1},PlotPoints\[Rule]1000],{n,0,10}]
Show[g]

What works to give an idea of what these  functions look like is:
Clear[norm, p0, p, g]
p0 = Table[LegendreP[n, x], {n, 0, 12}]
norm = Table[(1/Integrate[p0[[n]]*p0[[n]], {x, -1, 1}])^(1/2), {n, 1, 12}]
p = Table[p0[[n]]*norm[[n]], {n, 1, 12}]
g = Table[ParametricPlot3D[{p[[n + 1]]*p[[n + 2]], p[[n]]*p[[n + 2]],
    p[[n]]*p[[n + 1]]}, {x, -1, 1}, PlotPoints -> 1000], {n, 1, 10}]
Show[g]
Show[g, ViewPoint -> {3.354, -0.048, 0.447}]
Show[g, ViewPoint -> {0.045, 0.000, 3.383}]

The outcome seems to be two incoming and outgoing cyclinders.
That meet in a crosscap like figure?
They resemble Lerenz and Rössler chaotic trajectories.
I think these functions give an interesting set of new functions in 3D.

>  
>


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