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Derivation: Normalization condition of Legendre polynomials

  1. Feb 10, 2014 #1
    Greetings! :biggrin:

    1. The problem statement, all variables and given/known data
    Starting from the Rodrigues formula, derive the orthonormality condition for the Legendre polynomials:
    [itex] \int^{+1}_{-1} P_l(x)P_{l'}(x)dx=(\frac{2}{2l + 1}) δ_{ll'} [/itex]

    Hint: Use integration by parts

    2. Relevant equations
    [itex]P_l= \frac{1}{2^ll!}(\frac{d}{dx})^l (x^2-1)^l[/itex] (Rodrigues formula)
    ∫udv = uv -∫vdu (integration by parts)

    3. The attempt at a solution

    [itex] \int^{+1}_{-1} P_l(x)P_{l'}(x)dx = \frac{1}{2^{l+l'}l!l'!} \int^{+1}_{-1} (\frac{d}{dx})^l \,(x^2-1)^l \, (\frac{d}{dx})^{l'} \,(x^2-1)^{l'}\,dx [/itex]

    Integrating by parts:
    ∫udv = uv -∫vdu

    Let [itex] u = (\frac{d}{dx})^l (x^2-1)^l [/itex]
    [itex] \frac{du}{dx} = (\frac{d}{dx})^{l-1} (x^2-1)^{l-1} [/itex]
    [itex] du = (\frac{d}{dx})^{l-1} (x^2-1)^{l-1} dx[/itex]

    Let [itex]dv=(\frac{d}{dx})^{l'} (x^2-1)^{l'}dx [/itex]
    [itex] \int dv = \int (\frac{d}{dx})^{l'} (x^2-1)^{l'} dx[/itex]

    Question: How do I integrate [itex] \int (\frac{d}{dx})^{l'} (x^2-1)^{l'} dx[/itex] ?

    Thank you very much. :shy:
     
  2. jcsd
  3. Feb 10, 2014 #2

    vanhees71

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    2016 Award

    Think about the fundamental theorem of calculus
    [tex]\int \mathrm{d} x f'(x)=f(x)+C[/tex]
    for a function with a continuous derivative!
     
  4. Feb 11, 2014 #3
    Oooh... Yeah I remember. That was taught to us before. :biggrin: Thank you very much.
     
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