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Re: Q Legendre orthogonal polynomials mistake

  • To: mathgroup at smc.vnet.net
  • Subject: [mg67907] Re: Q Legendre orthogonal polynomials mistake
  • From: Roger Bagula <rlbagula at sbcglobal.net>
  • Date: Wed, 12 Jul 2006 05:06:13 -0400 (EDT)
  • 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

Paul Abbott,
Well it is nice to know it is built in.
My written documentation is older ( version3).
In that book only the P function is documented.
The Q was added in either version 4 or version 5.
Trying to run that code pulled down my system yesterday.( crash, burn... 
yeah)
My point was that my toral inverse polynomials
f[x_.n_]=x^n*LengendreP[n,1/x]
are new and different than LengendreQ[n,x].
I'd appreaciate you guys stop saying my questions are "nonsense."
I view some of the respomses as irresposible too.

Now that I know it is built in I can try a limited 1dimensional system using
the Q polynomials and see how that works.

See the post below that seems to have gotten lost!
Roger Bagula
Paul Abbott wrote:

 >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
 >School of Physics, M013                            Fax: +61 8 6488 1014
 >The University of Western Australia         (CRICOS Provider No 
00126G)   
 >AUSTRALIA                               http://physics.uwa.edu.au/~paul
 >
 >



Posted to that thread before but not up.

-------- Original Message --------
Subject: [mg67907] 	interesting non- orthogonal polynomials
From: 	Roger Bagula <rlbagula at sbcglobal.net>
To: mathgroup at smc.vnet.net
References: 	<e8nt4n$k7c$1 at smc.vnet.net> <e8qg8f$o63$1 at smc.vnet.net> 
<e8tbj5$227$1 at smc.vnet.net>



These inversions are probably a new set of polynomials?
The idea is to do the 1/x type inversion that
is invariant in tori to the Legendre polynomials.
The spherical harmonics that result aren't orthogonal to each other.
They are also non-orthogonal on  domain {1, Infinity}

They aren't the traditional Arctan[x] Q[n] functions associated with 
Legendre P{n]'s.
Mathematica doesn't have a function for those.
For the regular orthogonals the normalizaion is
Integrate[LegendreP[n, x]*LegendreP[m,x],{x,-1,1}]=delta[n,m]*2/(2*n+1)
http://www.du.edu/~jcalvert/math/legendre.htm
Most energy type calculastion are  made with the second type 
LegendreP[n,m,x]
tpolynomials that are used in Schrödinger calculations.
(* toral inverse polynomials*)
q0 = Table[ExpandAll[x^n*LegendreP[n, 1/x]], {n, 0, 5}]
normq = Table[(1/Integrate[q0[[n]]*q0[[n]], {x, -1, 1}])^(1/2), {n, 1, 6}]
q = Table[q0[[n]]*normq[[n]], {n, 1, 6}]
Inm = Table[N[Integrate[q[[n]]*q[[m]], {x, -1, 1}]], {n, 1, 6}, {m, 1, 6}]
MatrixForm[Inm]


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