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Problem Integrating a sequence of polynomials

  • To: mathgroup at smc.vnet.net
  • Subject: [mg124402] Problem Integrating a sequence of polynomials
  • From: Roger Bagula <roger.bagula at gmail.com>
  • Date: Wed, 18 Jan 2012 06:03:20 -0500 (EST)
  • Delivered-to: l-mathgroup@mail-archive0.wolfram.com

These polynomials are x^n+y^n+z^n=0
 Weierstrass-Beauville parametrics.
The results are Finsler space like minimal surfaces.
The z coordinate comes out to be Riemannian surface "sheet like"
and Mathematica really doesn't like my approach
to the problem.

Here is my code so far after about three or four attempts:
Clear[x, y, z, g, f]
x[n_] = If[Mod[n, 2] == 1, -f*(g^n + 1), I*f*(g^n + 1)]
y[n_] = f*(g^n - 1)
Table[Solve[x[n]^n + y[n]^n + z^n == 0, z], {n, 1, 5}]
w = {-(g^5 + 1), (g^5 - 1), (1 + 10  g^10 + 5  g^20)^(1/5)}
ParametricPlot3D[w, {g, -1, 1}]
ParametricPlot3D[w, {g, -10, 10}]
a = Table[-I*
   If[Mod[n, 2] == 1,
    w = {x[n]/f,
      y[n]/f, (z/f) /. Solve[x[n]^n + y[n]^n + z^n == 0, z][[1]]},
    w = {x[n]/f,
      y[n]/f, (z/f) /.
       Solve[x[n]^n + y[n]^n + z^n == 0, z][[1]]}], {n, 1, 5}]
z = r*Exp[i*t]
b = a /. g -> z /. f -> 1
Clear[x]
c = Table[Re[NIntegrate[b[[n]], {t, 0, x}]], {n, 1, Length[b]}]
Table[ParametricPlot3D[c[[n]], {r, 0, 1}, {x, 0, 2*Pi}], {n, 1,
  Length[c]}]



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