Q Legredre functions defined
- To: mathgroup at smc.vnet.net
- Subject: [mg67869] Q Legredre functions defined
- From: Roger Bagula <rlbagula at sbcglobal.net>
- Date: Tue, 11 Jul 2006 05:58:35 -0400 (EDT)
- References: <e8nt4n$k7c$1@smc.vnet.net> <e8qg8f$o63$1@smc.vnet.net> <e8tbj5$227$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
Defined very poorly in books I've seen(Jahnke and Emde, or Abramowitz
and Stegun)
Q[n]=ArcTan[x]*P[n]-W[n-1]
I haven't found a good formula for W[n]! (Jahnke and Emde. page 111)
W[n-1]=Sum[P[m-1]*P[n-m]/m,{m,1,n}]
In Mathematica notation:
W[x_,n_]=Sum[LegendreP[m-1,x]*LegendreP[n-m,x]/m,{m,1,n}]
Q[n_,x_]=ArcTan[x]*LegendreP[n],x-W[x,n]
I'm not having a lot of luck integrating these Q second kind
functions/ polynomials:
p0 = Table[Q[n, x], {n, 0, 5}]
norm = Table[(1/Integrate[p0[[n]]*p0[[n]], {x, -1, 1}])^(1/2), {n, 1, 6}]
p = Table[p0[[n]]*norm[[n]], {n, 1, 6}]
Inm = Table[N[Integrate[p[[n]]*p[[m]], {x, -1, 1}]], {n, 1, 6}, {m, 1, 6}]
MatrixForm[Inm]
These Q 's are the same as my toral inverse polynomials
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