Q Legendre orthogonal polynomials mistake
- To: mathgroup at smc.vnet.net
- Subject: [mg67877] Q Legendre orthogonal polynomials mistake
- From: Roger Bagula <rlbagula at sbcglobal.net>
- Date: Tue, 11 Jul 2006 05:58:51 -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
I got the ArcTan mixed with ArcTanh:
(*Jahnke and Emde, Page 111*)
W[x_, n_] = Sum[LegendreP[m - 1, x]*LegendreP[n - m, x]/m, {m, 1, n}]
Q[n_, x_] = ArcTanh[x]*LegendreP[n, x] - W[x, n]
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]
A way to check is the Integral (Abramowitz and Stegun page 337 gives):
Table[Integrate[LegendreP[n, x]* Q[m, x], {x, 1, Infinity}], {m, 0, 5},
{n, 0, 5}]
Table[(m-n)/(m+n+1), {m, 0, 5}, {n, 0, 5}]