1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Legendre Polynomial Integration

  1. Jul 6, 2017 #1

    joshmccraney

    User Avatar
    Gold Member

    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

    User Avatar
    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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted