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Homework Help: Spherical Harmonics

  1. Apr 14, 2006 #1


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    I'm trying to get the [itex]Y_l^l [/itex] spherical harmonic and I'm running into problems evaluating the following expression:

    \frac{d^{2l}(\cos^2(\theta) - 1)^l}{d\cos(\theta)^{2l}}

    The 2lth derivative with respect to cos theta of cos squared theta - 1 to the lth power
    it just seems like i'm going to end up with more and more terms as I keep differentiating..
    Last edited: Apr 14, 2006
  2. jcsd
  3. Apr 14, 2006 #2

    Tom Mattson

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    First of all, let's not make this more complicated then it has to be. Do a change of variables: [itex]x=\cos(\theta)[/itex].

    Now try to work it out for the first few values of [itex]l[/itex]. You should notice a simple pattern emerging. (Hint: I would expand the expression [itex](x^2-1)^l[/itex] before differentiating.
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