Show that int(-1,1) x^m P(x) dx = 0 if x < l where P(x) is legendres polynomials. The hint in the book says to use Rodrigues formula which is 1/2^ll! d^l/dx^l (x^2-1)^l
Ok it's clear that multiple integrations will result in 0 as (x^2-1)^l is in every integration and will always = 0.
1/2^ll! [(m!)(-1)^n int(-1,1) x^(m-n) ((x^2-1)^l d^l-n/dx^l-n]
I don't know how to show that that is 0. Eventually I know that since m < l that x^m will go away leaving just the (x^2-1)^l which is always 0 but I don't know a nice way of showing this. I also am not sure if that is a correct way of writing the above expression. n = the number of integrations that has happened. Any help would be appreciated thanks.
Re: legendres polynomials
I have not stepped though this, but it would seem you need to use Rodriques' formula and integrate repeatedly by parts, differentiating the power of x and integrating the derivative each time.