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Topic: Associated Legendre Function Problem in mma?
Replies: 1   Last Post: Jul 9, 1996 12:55 AM

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Posts: 72
Registered: 12/7/04
Associated Legendre Function Problem in mma?
Posted: Jul 8, 1996 1:56 AM
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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) /

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

0 1

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

2 -1

Sqrt[1 - x ]
3 Sqrt[-1 + x ]

4 1

Sqrt[1 - x ]
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@ee.stanford.edu

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