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It seems that the series expansion in powers of x^k that is given in the article yields the wrong sign for
Legendre polynomials of odd order. This can be checked by using this formula to calculate the lowest order polynomials by hand or with Mathematica. To obtain the correct formula, a factor (-1)^l
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If you define the
Legendre polynomials by Rodrigues' formula, then it is just Rolle's theorem, so a proof is not out of the question (but see my post below). That the roots are simple follows then from the fact that the polynomials satisfy a second order linear homogeneous differential equation, and
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Usually the singular makes more sense that the plural in the title of an article, but in this case, the title "Legendre polynomial" makes the same amount of sense as "Beattle" as the title of an article about John, Paul, George, and Ringo. The whole sequence
Legendre polynomials is to be thought of
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I have reverted the word "normalize" back to "standardize", since the polynomials are not normal, and the standardization is just a convention. I'm still not happy with this -- it's a conflict between saying exactly the right thing vs. belaboring a point that isn't really important. Improvements?
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I have added to the lead paragraph a summary of the proof of
Rodrigues' formula. However, I now wonder whether such a detailed proof benefits the article. We don't usually put detailed proofs in Knowledge articles, particularly when these proofs are fairly routine manipulations; I think taking
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In the section "Recursive definition", paragraph starting "Explicit representations include", the second formula involves binomial coefficients with negative (upper) argument. While the formula is not incorrect, it would be more elegant to express in terms of binomial coefficients with positive
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in a too narrow sense. Requiring P_n(1)=1 is a true normalization, just not w.r.t. the
Hilbert space norm, but the sup-norm (L_\infty-normalization). In the same spirit, the property that Legendre polynomials take the maximal value in x=1 is of independent interest and deserves to be mentioned
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is the correct word here. Normalization does not necessarily mean to transform something to norm one, it is used for any transformation to get something in standard form. However, it is not that important. I do wonder though why the word "standardize" is written in bold. --
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convergence to a piecewise continuous function, that takes more work. The person who wrote those words doesn't seem aware of the difference kinds of convergence or of the fact that mathematicians may often construe "linear combination" to mean
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Works for me. Nothing wrong with adding a sentance that says: "Sometimes this is called a "normalization", although the correct meaning of normalization is that the integral of the square of the function should be unity."
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The
Legendre polynomials form an orthogonal basis for the Hilbert space of square-integrable functions on the interval from -1 to 1, so we're talking about a "linear combination" in the Hilbert-space sense, i.e.,
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so cannot vanish to higher order at any point of their domain, by uniqueness of solutions. Much more can be said about the zeros of the
Legendre polynomials, and there should be a section about them.
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in section
Rodrigues' formula and other explicit formula, second equation, is undefined. I would guess it means the Taylor coefficient of t^n, but to my knowledge, this is not universal notation.
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I am changing "generating function" to "ordinary generating function". I also think the word "Taylor" is incorrect and should be removed; which I will do in about a week unless somebody objects.
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in a distributional form, but no one dares to talk about the (far easier formulated) convergence in the
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Can anyone provide more concrete examples of applications of the
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Yes, well, Hilbert spaces should be discussed in the article on orthogonal polynomials anyway, so as to cover the whole kit-n-kaboodle.
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The roots of the Legendre polynomials are very important, e.g. for Gaussian quadrature. There are a number of beautiful proofs that
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When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.
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Strange that in the same section you would talk about pointwise convergence towards piecewise continuous functions, and state the
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would have to be included. There is no reference given for this formula, and it does not appear in either Abramowitz or Arfken.
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are convolutions of the Legendre polynomials, which is an interesting and unexpected relation—although it could use a source.
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The very last formula in that section is however different—it seems to say that the Chebyshev polynomials of the second kind
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Also, is it correct that P_n solves the differential equation with parameter n, or is n used in two different senses here?
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speak of a "Legendre polynomial" in the singular in some contexts, but generally those are of interest only because this
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These can't be bases and linear combinations in the sense of linear algebra. Are we talking Hilbert space basis here?
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for the space of piecewise continuous functions defined on this interval, so any such function can be written as a
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https://web.archive.org/web/20090920070434/http://edocs.ub.uni-frankfurt.de/volltexte/2007/3757/pdf/A009566090.pdf
1146:. It would be nice to mention this property (perhaps with a proof or at least with a reference) in the article.
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to delete these "External links modified" talk page sections if they want to de-clutter talk pages, but see the
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arguments. This is easy to do. I propose to re-express this formula in this way, unless there are objections.
907:{\displaystyle f_{lm}(\theta )={\frac {(\sin \theta )^{|m|}}{2^{l}l!}}\left^{l+|m|}(\cos ^{2}(\theta )-1)^{l}}
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If you have discovered URLs which were erroneously considered dead by the bot, you can report them with
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I think that for special functions explanations of some easy (quick) proofs in the article makes sense.
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consists mostly of a big table, which as far as I can tell is just giving the coefficients for the
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on Knowledge. If you would like to participate, please visit the project page, where you can join
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on Knowledge. If you would like to participate, please visit the project page, where you can join
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before doing mass systematic removals. This message is updated dynamically through the template
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580:{\displaystyle \lim _{n\rightarrow \infty }\int _{-1}^{1}\left|s_{n}(x)-f(x)\right|^{2}\,dx=0.}
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I feel that they are as important as orthogonality relations yet receive little mention.
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https://web.archive.org/web/20060427014500/http://www.du.edu/~jcalvert/math/legendre.htm
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If you found an error with any archives or the URLs themselves, you can fix them with
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381:{\displaystyle \sum _{n=0}^{\infty }{\frac {2n+1}{2}}P_{n}(x)P_{n}(y)=\delta (x-y)}
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http://edocs.ub.uni-frankfurt.de/volltexte/2007/3757/pdf/A009566090.pdf
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Is it appropriate to add the completeness relation to this article?
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One cannot just take finite angles an expect the identity to work.
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The last two identities here are wrong. The correct limit is
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My quantum mechanics text says speaks of something called the
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Not everything verifiable is worth including in an article.
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Is this something that somehow missed having an article? --
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for additional information. I made the following changes:
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No, you just didn't read the article carefully enough.
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1829:Start-Class physics articles of Mid-importance
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137:Mid-priority
136:
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62:Mid‑priority
40:WikiProjects
1564:Now done.
686:The Beatles
644:as a unit.
112:Mathematics
103:mathematics
59:Mathematics
30:Start-class
1803:Categories
1700:Report bug
1237:Rrogers314
1010:normalized
1683:this tool
1676:this tool
1111:roots in
592:pointwise
442:AxelBoldt
1689:Cheers.—
1580:contribs
1568:unsigned
1527:unsigned
1500:unsigned
1351:catslash
1328:Kkddkkdd
1245:contribs
1233:unsigned
1044:unsigned
681:Beatles'
624:unsigned
1603:my edit
1572:Uhap027
1552:Uhap027
464:is the
244:on the
217:Physics
208:Physics
164:Physics
139:on the
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1347:degree
1324:degree
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678:, but
675:Beatle
597:finite
455:finite
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1343:order
1320:order
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1210:Ulner
997:linas
970:linas
919:Smack
609:linas
426:basis
1791:talk
1776:edit
1761:talk
1576:talk
1556:talk
1535:talk
1508:talk
1355:talk
1332:talk
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1199:talk
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