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Legendre Polynomial Integration

  1. Jul 6, 2017 #1

    joshmccraney

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    1. The problem statement, all variables and given/known data
    Simplify $$\int_{-1}^1\left( (1-x^2)P_i''-2xP'_i+2P_i\right)P_j\,dx$$
    where ##P_i## is the ##i^{th}## Legendre Polynomial, a function of ##x##.
    2. Relevant equations

    3. The attempt at a solution
    Integration by parts is likely useful?? Also I know the Legendre Polynomials are orthogonal on ##[-1,1]##. Before simply trying integration by parts term-wise I was trying to see if the equi-dimensional equation ##(1-x^2)P_i''-2xP'_i+2P_i## could first be expressed in a more simple manner? Any thoughts?
     
  2. jcsd
  3. Jul 6, 2017 #2

    joshmccraney

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    Gold Member

    Just realized ##(1-x^2)P_i''-2xP'_i = ((1-x^2)P'_i)'##, and obviously the last term ##2P_i## instantly vanishes when ##i\neq j##. Then the above integral becomes $$(1-x^2)P'_iP_k|_{-1}^1-\int_{-1}^1(1-x^2)P'_iP'_k\,dx+\frac{2}{2i+1}\delta_{ik}$$ where I use the kronicker delta. Does this simplify further?

    Edit: yes it does further simplify. Look at Legendre's DE. It's fairly straightforward from here on out.
     
    Last edited: Jul 6, 2017
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