Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Please help integration using rodrigues formula.

  1. Jun 5, 2010 #1
    The book gave the integration of a function with the legendre polynomial formula:

    [tex]\int_{-1}^1 f(x)P_n(x)dx = \frac{1}{2^n n!}\int_{-1}^{1} f^n(x)(x^2-1)^n dx[/tex]

    It just said the formula can be obtained by repeat using Rodrigues formula and integral by parts but did not go into detail. I want to work out the steps and I got stuck. This is what I have:

    Using Rodrigues:

    [tex]\int_{-1}^{1} f(x)P_n(x)dx = \frac{1}{2^n n!}\int_{-1}^{1} f(x)\frac{d^n}{dx^n}[(x^2-1)^n] dx[/tex]

    After the first integral by parts:

    [tex]\frac{1}{2^n n!}\int_{-1}^{1} f(x)\frac{d^n}{dx^n}[(x^2-1)^n] dx = \frac{1}{2^n n!}[f(x)\frac{d^{n-1}}{dx^{n-1}}[(x^2-1)^n]]_{-1}^1 \;-\; \frac{1}{2^n n!}\int_{-1}^{1} f^{(1)}(x)\frac{d^{n-1}}{dx^{n-1}}[(x^2-1)^n] dx[/tex]

    In order for this to continue to the next integration by parts, the following has to be true:

    [tex]\frac{1}{2^n n!}[f(x)\frac{d^{n-1}}{dx^{n-1}}[(x^2-1)^n]]_{-1}^1 = 0[/tex]

    [tex]\Rightarrow\; [\frac{d^{n-1}}{dx^{n-1}}[(x^2-1)^n]]_{-1}^1 = 0[/tex]

    I don't know whether my assumption is correct. If so, I still don't know how it is equal to zero. can anyone give me some guidance.
    Last edited: Jun 5, 2010
  2. jcsd
  3. Jun 5, 2010 #2


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    I haven't checked you work, but assuming it is correct, this step:

    \Rightarrow\; [\frac{d^{n-1}}{dx^{n-1}}[(x^2-1)^n]]_{-1}^1 = 0

    can be written as

    [\frac{d^{n-1}}{dx^{n-1}}[(x+1)^n(x-1)^n]]_{-1}^1 = 0

    Using the product rule differentiating n-1 times I think you will find at least a factor of (x-1)(x+1) in every term of the expansion. That would give you the answer 0.
  4. Jun 6, 2010 #3
    Thanks for you help.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook