Re: Q Legendre orthogonal polynomials mistake
- To: mathgroup at smc.vnet.net
- Subject: [mg67904] Re: Q Legendre orthogonal polynomials mistake
- From: Paul Abbott <paul at physics.uwa.edu.au>
- Date: Wed, 12 Jul 2006 05:06:06 -0400 (EDT)
- Organization: The University of Western Australia
- References: <e8nt4n$k7c$1@smc.vnet.net> <e8qg8f$o63$1@smc.vnet.net> <e8tbj5$227$1@smc.vnet.net> <e8vtqe$ssc$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
In article <e8vtqe$ssc$1 at smc.vnet.net>,
Roger Bagula <rlbagula at sbcglobal.net> wrote:
> 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}]
I do not understand your question. First, note that LegendreQ is
built-in. Next, note that there are 3 types of Legendre function (the
3rd type is required here) and that the integral on Abramowitz and
Stegun page 337, which is online at
http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP
is only valid with conditions on the indices.
As a check of Abramowitz and Stegun 8.14.1,
Table[1/((m - n) (m + n + 1)) ==
Integrate[LegendreP[n, x]*LegendreQ[m, 0, 3, x], {x, 1, Infinity}],
{n, 0, 5}, {m, n + 1, 5}] // Flatten // Union
{True}
Cheers,
Paul
_______________________________________________________________________
Paul Abbott Phone: 61 8 6488 2734
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