Orthogonality of Legendre Polynomials

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Homework Statement


For spherical coordinates, we will need to use Legendre Polynomials,
a.Sketch graphs of the first 3 – P0(x), P1(x), and P2(x).

b.Evaluate the orthogonality relationship (eq 3.68) to show these 3 functions are
orthogonal to each other. (3 integrals).

c.Show that the normalization result is 2/2l+1 as stated in eq 3.68.


Homework Equations





The Attempt at a Solution


I believe I got part a, but I am unsure how to do the rest. I don't see where the equation that has the theta terms, comes into use.
 

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The equation with the theta terms is irrelevant. It is there to show the main application, namely Spherical Harmonics. You just need to carry out the integration to show you get 0 for distinct l,l' and 1 (with the given normalization) if l = l'.
 
Ahh ok I was wondering this. I realized maybe this was the case a few minutes ago, and tried to work it out. Here is what I just did. I think this is correct... Any thoughts are appreciated.
 

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