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AES
Posts:
72
Registered:
12/7/04


Associated Legendre Function Problem in mma?
Posted:
Jul 8, 1996 1:56 AM


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^21)^{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^21)^(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@ee.stanford.edu



