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Thread: legendres polynomials

  1. #1
    Junior Member
    Join Date
    Oct 2009
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    New York
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    65

    legendres polynomials

    Show that int(-1,1) x^m P(x) dx = 0 if x < l where P(x) is legendres polynomials. The hint in the book says to use Rodrigues formula which is 1/2^ll! d^l/dx^l (x^2-1)^l

    Ok it's clear that multiple integrations will result in 0 as (x^2-1)^l is in every integration and will always = 0.

    This leaves

    1/2^ll! [(m!)(-1)^n int(-1,1) x^(m-n) ((x^2-1)^l d^l-n/dx^l-n]

    I don't know how to show that that is 0. Eventually I know that since m < l that x^m will go away leaving just the (x^2-1)^l which is always 0 but I don't know a nice way of showing this. I also am not sure if that is a correct way of writing the above expression. n = the number of integrations that has happened. Any help would be appreciated thanks.

  2. #2
    Elite Member
    Join Date
    Sep 2005
    Posts
    7,294

    Re: legendres polynomials

    I have not stepped though this, but it would seem you need to use Rodriques' formula and integrate repeatedly by parts, differentiating the power of x and integrating the derivative each time.

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