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I Legendre's equation

  1. Jul 27, 2017 #1
    Hi
    I want to show that ##V(x) = (1-x^2)^{m/2} P_l (x)## is a solution of the equation
    $$\frac{d}{dx} \bigg[(1-x^2) \frac{dV(x)}{dx} \bigg] + \bigg[l(l+1) - \frac{m^2}{1-x^2} \bigg]V(x) = 0.$$ Because the equation for ##P_l (x)## is $$\frac{d}{dx} \bigg[(1-x^2) \frac{dP_l (x)}{dx}\bigg] + l(l+1) P_l (x) = 0,$$ my attempts have been consisting of trying to differentiate the last equation ##m## times. I did not realize how to use the Leibnitz formula for general derivative, so I have been trying to differentiate it a small number of times and by induction arguing what it looks like after the ##m \text{-th}## derivative. I'm having no success in by proceeding this way, though.

    However, this is just an attempt. I would like equally well any other one, because I just want to show that ##V(x)## given as above is a solution for the equation shown.
     
    Last edited: Jul 27, 2017
  2. jcsd
  3. Jul 30, 2017 #2

    Charles Link

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    A google of the subject shows that ## V(x)=(-1)^m (1-x^2)^{m/2} \frac{d^m P_l(x)}{dx^m} ##, so that I believe the ## V(x) ## as you presented it is incorrect. ## \\ ## Editing: I think your solution for ## P_l(x) ## and its associated equation is simply the case where ## m=0 ##, but then you don't need the ## (1-x^2)^{m/2} ## term in ## V(x) ##, and the result of the ## P_l(x) ## equation follows immediately.
     
    Last edited: Jul 30, 2017
  4. Jul 30, 2017 #3
    I think you misunderstood my question. I have actually showed what the solution is, and I said I was interested in showing that that expression (I missed the ##(-1)^m## term) for ##V(x)## is a solution for the equation.

    It turns out that after starting this thread I was able to derive both the Legendre's polynomials and the associated Legendre equation (and its solution) with the help of the manuscript on math-phys by @vanhees71. :smile:
     
  5. Aug 3, 2017 #4

    Svein

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    It looks to me like a Sturm-Liouville equation...
     
    Last edited: Aug 3, 2017
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