The Math Forum

Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Math Forum » Discussions » Inactive » comp.soft-sys.math.mathematica

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

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

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]

Posts: 72
Registered: 12/7/04
Associated Legendre Function Problem in mma?
Posted: Jul 8, 1996 1:56 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

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.


Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© The Math Forum at NCTM 1994-2018. All Rights Reserved.