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Legendre polynomial property

  1. May 5, 2007 #1
    1. The problem statement, all variables and given/known data

    I am to prove that P_n(-x)=(-1)^n*P_n(x)

    And, P'_n(-x)=(-1)^(n+1)*P'_n(x)

    2. Relevant equations

    3. The attempt at a solution

    I know that whether a Legendre Polynomial is an even or odd function depends on its degree.It follows directly from the solution of Legendre differential equation.But,to prove these properties I am getting stuck.

    Can anyone help me to start with these?
  2. jcsd
  3. May 5, 2007 #2


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    Staff: Mentor

  4. May 5, 2007 #3
    I referred to those webpages earlier.They contain results but no derivation.

    How can I use (-x)^a=(-1)^a*(x)^a within a series?
  5. May 6, 2007 #4


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    Gold Member

    As Astronuc suggested, substitute as (-x)a for each term in the series. What can you say about (-1)n-2m when m varies?
  6. May 6, 2007 #5
    OK,I already got it.
    One should exploit the property: P_n(x) even and odd according as n is even and odd.
  7. May 6, 2007 #6

    The statement you are going to prove can be seen readily if you look at the Ridrigues' formula of Lengreda Polynomial, isn't it? :wink:
  8. May 6, 2007 #7
    I am interested to know how the first formula can be derived from Rodrigues's formula.Is the Replacing x by -x should affect (d^l/dx^l)?If so,how?

    Actually it appears...but not quite sure,how?
  9. May 6, 2007 #8
    just change x-> -x
    then the Rodrigues's formula has d/d(-x)
    which can be rewritten as dx/d(-x)*d/d(x) , like changing variables as usual
    then d/d(-x) = -d/dx and d^n/d(-x)^n = (-1)^n d^n/dx^n

    for the question how the Rodrigues's formula came from...
    i have no idea at all...:confused:
  10. May 6, 2007 #9
    I anticipated something like this.Thanks for clarification.
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