Talk:Associated Legendre polynomials

message: There are two sign conventions for associated Legendre polynomials; some authors include a factor of \left(-1\right)^{m} (Condon-Shortley phase) (e.g. Arfken, Mathematical methods for physicists, p.669). For Geophysicists it is interesting to know that Grant and West (Interpretation Theory in Applied Geophysics, 1965, p.223) omit the factor. See also http://mathworld.wolfram.com/Condon-ShortleyPhase.html  —Preceding unsigned comment added by 124.177.116.210 (talk) 00:40, 31 October 2009 (UTC)
thread_title: Sign Convention

message: There are 3 aspects of my recent changes that might warrant
discussion or controversy:
(1) I have used the superscript-parameters-in-parentheses convention,
the same as was done in the orthogonal polynomials page.  This was
inspired by the convention used with the Jacobi and Gegenbauer
polynomials, but is not "industry standard" beyond that.  I'm
beginning to wonder whether it's worth it.  This isn't "original
research", but it might border on "original standardization".
(2) I have taken out the implication that P_\ell^m with
negative values of m is actually meaningful.  It's true that there is
a very important notion, in the field of spherical harmonics, that m
lies in the range -\ell \le m \le \ell.  (The periodic
table of the elements would look very different if this were not so!)
But those negative values don't specify things like P_4^{-3}.
They specify a different phase of the polar angle \phi.
Therefore, I believe that, prior to making the jump from Legendre
polynomials to spherical harmonics and quantum mechanics, the requirement
should be 0 \le m \le \ell.
(3) Some texts (Abramowitz and Stegun, Whittaker and Watson) define things
with (z^2-1)^{\mu/2} instead of the (1-z^2)^{\mu/2}
used here.  That would be more natural when one is defining analytic functions
on the complex plane, but doesn't work for the important case of the real
interval [-1, 1].  Accordingly, I have switched things, and changed and checked
the formulas.  When generalized to the complex plane, things work out the
same as in A&S and W&W.
Comments?  Complaints?
William Ackerman 20:42, 12 May 2006 (UTC)
thread_title:  My recent changes 

message: Note that the correct name for these functions are Associated Legendre Functions -- they are not always polynomials. A bit of a change just to do without discussion. Thoughts? MFago 23:08, 1 June 2006 (UTC)
replies: text: We should not care much what the correct name is, but we should use the name actually being used. I checked Abramowitz & Stegun and they use Associated Legendre function. That supports changing the name. -- Jitse Niesen (talk) 14:25, 2 June 2006 (UTC)

text: I'm not entirely certain that "associated Legendre polynomials" is more prelevant in the literature, but that, as you suggest, the existing nomenclature is messed up. How about (since Associated Legendre functions redirects here) adding a single sentence to the introduction similar to The associated Legendre polynomials are also referred to as associated Legendre functions which is perhaps more correct as they are not polynomials when m\neq0. MFago 14:51, 13 June 2006 (UTC)
replies: text: In fact, there is just such a sentence in "Definition" -- I'd suggest simply moving it into the introduction. MFago 20:23, 13 June 2006 (UTC)

text: Of course.  When there is no good way of presenting it which conceals the facts, just lay out the facts.  Done.  I also changed a section title to avoid using the phrase "general Legendre functions". William Ackerman 22:17, 13 June 2006 (UTC)
thread_title: Not polynomial

message: Associated Legendre polynomials → Associated Legendre function — As noted by others, these functions are not generally polynomials. The page probably would have been moved already but it can't be, because the target name already exists and redirects to Associated Legendre polynomials. I think we should go with the term function, and let A.L. polynomials redirect to it. EricK 18:01, 21 February 2007 (UTC)
replies: text: Add  # Support  or  # Oppose  on a new line in the appropriate section followed by a brief explanation, then sign your opinion using ~~~~.  Please remember that this survey is not a vote, and please provide an explanation for your recommendation.
====Survey - in support of the move====
#Support: The functions are simply not always polynomials, and this leads to lots of confusion. As an example, Gauss-Legendre quadrature integration schemes are only exact when the integrand is a polynomial, but if the integrand were to be a Legendre function, all bets are off. Lunokhod 19:59, 21 February 2007 (UTC)

text: Add any additional comments:
"The differential equation is also invariant under a change from ℓ to −ℓ − 1, and the functions for negative ℓ are defined by
    P_{-\ell} ^{m} = P_{\ell-1} ^{m},\ (\ell=1,\,2,\, ...). "
This sentence and the formula under it are inconsistent!  — Preceding unsigned comment added by 144.122.30.156 (talk) 10:41, 25 September 2013 (UTC)
thread_title:  Requested move 

message: :Add  # Support  or  # Oppose  on a new line in the appropriate section followed by a brief explanation, then sign your opinion using ~~~~.  Please remember that this survey is not a vote, and please provide an explanation for your recommendation.
====Survey - in support of the move====
#Support: The functions are simply not always polynomials, and this leads to lots of confusion. As an example, Gauss-Legendre quadrature integration schemes are only exact when the integrand is a polynomial, but if the integrand were to be a Legendre function, all bets are off. Lunokhod 19:59, 21 February 2007 (UTC)
thread_title: Survey

message: #Support: The functions are simply not always polynomials, and this leads to lots of confusion. As an example, Gauss-Legendre quadrature integration schemes are only exact when the integrand is a polynomial, but if the integrand were to be a Legendre function, all bets are off. Lunokhod 19:59, 21 February 2007 (UTC)
thread_title: Survey - in support of the move

message: :Add any additional comments:
"The differential equation is also invariant under a change from ℓ to −ℓ − 1, and the functions for negative ℓ are defined by
    P_{-\ell} ^{m} = P_{\ell-1} ^{m},\ (\ell=1,\,2,\, ...). "
This sentence and the formula under it are inconsistent!  — Preceding unsigned comment added by 144.122.30.156 (talk) 10:41, 25 September 2013 (UTC)
thread_title: Discussion

message: Why do I get automaticly redirected to the site with the Associated Legendre functions when i search for the page on the Legendre Functions? They are obviously not identical. It would be enlightening to have a page on the general case also. Neltah 15:45, 11 May 2007 (UTC)
thread_title:  Integer variables? 

message: Nobody writes brackets (parentheses) around the order.  They need to be removed.  Brackets (parentheses) are reserved for the Jacobi polynomials and they are not used for associated Legendre functions in the literature I have seen. HowiAuckland 21:59, 16 September 2007 (UTC)
replies: text: The parentheses surprised me; I've not seen them elsewhere and a quick trawl through Google books and my own bookshelves threw up zero instances of this notation. Unless they are supported by the references, the paranthesis should go. --catslash (talk) 00:49, 10 August 2009 (UTC)

text: ...and I shall remove them if nobody leaps to their defence within the week --catslash (talk) 16:12, 11 August 2009 (UTC)

text: (Done)--catslash (talk) 09:59, 22 August 2009 (UTC)
thread_title:  Extremely bad notation 

message: The beginning of this article should refer to the most general form of associated Legendre function, i.e. those with complex order, degree and argument.  These are important special functions and are used to specifically describe spherical harmonics, oblate and prolate spheroidal harmonics, conical functions, toroidal harmonics, etc.  All of these require different choices (and sometimes complex valued) order, degree and argument.  Also, the argument can be real valued but simply greater than unity.  Furthermore there are many formulae for associated Legendre functions which can be written most easily and in the most generality using general complex valued order and degree.  There are many examples of these and those of you who are familiar with Abramowitz and Stegun are well aware. In order for this article to be the most general, this should be done.  If somebody wants to start a separate article which just works for those associated Legendre functions which are used in spherical harmonics, i.e. those with integer degree and order, and with real argument between plus and minus one, this should be performed.  However, this article should be applied to the most general case, otherwise this article should be renamed. HowiAuckland 02:10, 24 September 2007 (UTC)
thread_title:  Complex Generality 

message: In the definition of p, note that n-m-u is always nonpositive, so p=0. Therefore something is wrong with the formula.  —Preceding unsigned comment added by 129.88.33.185 (talk) 09:40, 1 October 2009 (UTC)
thread_title:  Gaunt formula 

message: I found the following posted on my personal talk page; I think it's better to reply here. linas (talk) 21:32, 13 November 2010 (UTC)
replies: text: I do not understand you idea : . As such, Legendre polynomials can be generalized (In what way?)  to express the symmetries of semi-simple Lie groups (not SO(3)?)  and Riemannian symmetric spaces. (not euclidic ?) Gvozdet (talk) 12:02, 11 October 2010 (UTC)
replies: text: Hi, Sorry for the late response, I do not get on Wikipedia very often. This is not "my idea"; rather, there are papers, books and conferences on the topic. Simply-put, whenever one has a space with a continuous symmetry, one also has a symmetry group describing the symmetry; these are essentially the Lie groups. One may define a Laplacian on such spaces, and then study the eigenfunctions of the Laplacian on these spaces. These solutions can be considered to result in "generalizations" of the associated Legendre polynomials. The bottom of the article on spherical harmonics provides more detail, and at least four references. linas (talk) 21:32, 13 November 2010 (UTC)
thread_title:  Generalizations 

message: I believe at least some of the recurrence formulae were copied from a source that excludes the Condon-Shortley phase (which the rest of this article includes). For example I tested the second relation on Wolfram Alpha here and got the negative of what the article says it should be.
I will correct the ones I think are wrong and add another recurrence relation I derived from the others and found useful. However since I don't have a reference (e.g. Abramowitz and Stegun write about Legendre functions of a complex variable and it's subtly different) and I haven't derived them myself, I can't be 100% sure of their accuracy.
Edit: This book{{cite book|last=Attar|first=Rafaat A. El|title=Special functions and orthogonal polynomials|year=2006|publisher=Lulu Press|location=Morrisville, N.C.|isbn=1411666909}|pages=164} has recurrence relations equivalent to some in the article but defines the polynomials without the Condon-Shortley factor.
202.8.37.86 (talk) 05:57, 27 January 2012 (UTC)
thread_title:  Recurrence relations 

message: "This equation has nonzero solutions that are nonsingular on [−1, 1] only if ℓ and m are integers with 0 ≤ m ≤ ℓ, or with trivially equivalent negative values." Where is this statement taken from? I have not been able to find it in the literature. 5.61.176.43 (talk) 00:07, 9 April 2016 (UTC)
thread_title:  Reference needed 

message: Substituted at 21:34, 26 June 2016 (UTC)
thread_title: Assessment comment