Legendre Polynomial Orthogonality Integral Limits

[sir_manning]

Good afternoon

I have a question regarding the limits on the orthogonality integral of Legendre Polynomicals:

$$\int_{-1}^1 P_l(u)P_{l'}du = 2/(2l+1)$$

I am in the middle of a question involving the solution of Laplace's equation inside a hemisphere, which means that for the usual $$u=cos\theta$$, the limits will be from 0 to 1 instead. So after solving for my boundary conditions and multiplying by $$P_{l'}$$, I have:

$$P_{l'}(u)C = A_{l}\sum{P_{l}(u)P_{l'}(u)}$$

The next step is to integrate each side. I want to use the orthogonality integral above, but since I have to integrate the LHS from 0 to 1, won't I have to do the same for the RHS? Or can I integrate from -1 to 1 on each side, and then change the limit on the LHS because it isn't defined for 0 to 1. I've already done the latter and the result seems to work out, but I'm simply wondering if it is justified or not.

 

[kurt.physics]

Thank you in advance for your help.Yes, it is justified to integrate from -1 to 1 on each side and then change the limit on the left hand side because it isn't defined for 0 to 1. This is a valid approach and will yield the correct result.