Homework Help: Messy integral

1. Oct 1, 2006

Logarythmic

Problem:

Show that

$$\int_{-1}^{1} x P_n(x) P_m(x) dx = \frac{2(n+1)}{(2n+1)(2n+3)}\delta_{m,n+1} + \frac{2n}{(2n+1)(2n-1)}\delta_{m,n-1}$$

I guess I should use orthogonality with the Legendre polynomials, but if I integrate by parts to get rid of the x my integral equals zero.
Any tip on how to start working with this?

2. Oct 1, 2006

Astronuc

Staff Emeritus
First thought would be to use one of the recursion relationships on xPn(x).

For example -

$$(l+1)P_{l+1}(x)\,-\,(2l+1)xP_l(x)\,+\,lP_{l-1}(x)\,=\,0$$

BTW, has one shown -

$$\int_{-1}^{1} P_n(x) P_m(x) dx = \frac{2}{2n+1}\delta_{m,n}$$

That was demonstrated here on PF recently.

Last edited: Oct 1, 2006
3. Oct 1, 2006

Logarythmic

Yes, I've got the last equation and I'll try with the recursion, thank you. =)

4. Oct 1, 2006

StatMechGuy

Another thing I would recommend is to try using the Rodriguez formula for the Legendre polynomials, then play games with integration by parts.

5. Oct 1, 2006

Logarythmic

And why is that? I solved the problem by the way. Pretty simple when you know about the recursion relationships.