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Proving Rodrigues' formula

  1. Oct 12, 2011 #1
    Hello! Refering to the note:

    http://www.serc.iisc.ernet.in/~amohanty/SE288/l.pdf" [Broken]

    a definition of the legendre polynomials is

    [tex]P_n(\cos \theta) = \frac{(-1)^n}{n!} r^{n+1} \frac{\partial^n}{\partial z^n}\left(\frac{1}{r} \right)[/tex]

    I want to show that this is equivalent with the more familiar Rodrigues' formula

    [tex] P_n(x) = \frac{1}{2^n n!} \frac{d^n}{dx^n} \left[(x-1)^n\right][/tex]

    where [tex]x = \cos \theta[/tex]. I've tried various things such as replacing ddz in terms of polar coordinates, but I realize that I'm walking blind. Idealy I would want to go from the first relation in terms of partials with respect to z to Rodrigues' formula.
    Last edited by a moderator: May 5, 2017
  2. jcsd
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