Integrating Legendre Polynomials Pl & Pm

[Elliptic]

Homework Statement

Integrate the expression
Pl and Pm are Legendre polynomials

Homework Equations

The Attempt at a Solution

Suppose that solution is equal to zero.

 

[TheFurryGoat]

What properties do you know about Legendre Polynomials? If you can use the orthogonal properties that are listed in the article on Legendre polynomials in wikipedia, then integration by parts should do the trick.

 

[Elliptic]

What properties do you know about Legendre Polynomials? If you can use the orthogonal properties that are listed in the article on Legendre polynomials in wikipedia, then integration by parts should do the trick.

But, how make Pm'(x) I don't understand(recurrent differentiation formula?)

 

[TheFurryGoat]

Under the orthogonality section in the wikipedia article on Legendre polynomials, you find the identity
$\displaystyle \frac{d}{dx}\left[(1-x^2)\frac{d}{dx}P(x)\right] = -\lambda P(x)$
where the eigenvalue $\lambda$ corresponds to $n(n+1).$
I suppose $P(x)=P_n(x)$ for any $n$, but I'm not sure though. If this is the case, and you know this property, then integration by parts should do the trick.

 

[Elliptic]

Thanks for help, I succeeded to do job.