# Proof - Legendre polynomials

1. Feb 10, 2012

### JiriV

Following relation seems to hold:

$\int^{1}_{-1}\left(\sum \frac{b_{j}}{\sqrt{1-μ^{2}}} \frac{∂P_{j}(μ)}{∂μ}\right)^{2} dμ = 2\sum \frac{j(j+1)}{2j+1} b^{2}_{j}$

the sums are for j=0 to N and $P_{j}(μ)$ is a Legendre polynomial. I have tested this empirically and it seems correct.

Anyway, I would like to have i) either a proof, or ii) a reference in a book, by which it is obtained easily. Do you have some suggestions?

Thank you.

2. Feb 10, 2012

### chiro

Hey JiriV and welcome to the forums.

I'm not too familiar with the legendre polynomial myself but I think I might be able to offer a few suggestions.

One suggestion is to expand the results of the integral and simplify. Another suggestion is to use the properties of the projection via an integral transform and a correct basis. For more of the specifics on this check out the following link and scroll down to the Gram-Schmidt process for getting the bases:

http://mathworld.wolfram.com/LegendrePolynomial.html

You might actually be better off doing the projection or using properties of the derivative in conjunction with the projection, but you would have to do a bit of investigation on your part.

3. Feb 17, 2012