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MathGroup Archive 1996

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Associated Legendre Function Problem in mma?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg4361] Associated Legendre Function Problem in mma?
  • From: siegman at ee.stanford.edu (A. E. Siegman)
  • Date: Mon, 15 Jul 1996 05:01:00 -0400
  • Organization: Stanford University
  • Sender: owner-wri-mathgroup at wolfram.com

Associated Legendre functions are a real bear.  Trying to 
cope with them, I note in Abramowitz and Stegun, p. 334, 
Eq. (8.6.16), that one of these functions which I 
particularly need to use, LegendreP[n,-n,x], has the special 
case (in TeX notation):

   P_n^{(-n}(x) = 2^{-n} (x^2-1)^{n/2} / \Gamma[n+1]

But when I try to confirm this with mma, I see that the
magnitudes are OK, but there is a residual confusion
about phase angles:

specialCase[n_,x_] := 2^(-n) (x^2-1)^(n/2) /
                      Gamma[n+1]

Table[{n, LegendreP[n,-n,x] /
          specialCase[n,x] // Simplify},
      {n,0,5}]  // TableForm

0   1

              2
    Sqrt[1 - x ]
    -------------
               2
1   Sqrt[-1 + x ]

2   -1

                2
      Sqrt[1 - x ]
    -(-------------)
                 2
3     Sqrt[-1 + x ]

4   1

              2
    Sqrt[1 - x ]
    -------------
               2
5   Sqrt[-1 + x ]

and unfortunately getting the phase angles right is 
important in my problem.  Who's correct here?

Addendum:  The reason for worrying about this is that
I want to evaluate very high order polynomicals (n > 50)
using rational fraction values of x for accuracy (which 
seem to work pretty well).  But while LegendreP[2n, x],
which I also need to use, seems to run fine in this way,
LegendreP[n,-n,x] slows to a crawl for n > 20 or thereabouts
-- even though the polynomial expressions for the regular
and associated Legendre's are of the same order in the
two cases.  Hence the search for an alternative for the
associated case.

   --AES   siegman at ee.stanford.edu

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