Good afternoon(adsbygoogle = window.adsbygoogle || []).push({});

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

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

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

[tex]P_{l'}(u)C = A_{l}\sum{P_{l}(u)P_{l'}(u)}[/tex]

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.

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Legendre Polynomial Orthogonality Integral Limits

Can you offer guidance or do you also need help?

**Physics Forums | Science Articles, Homework Help, Discussion**