How Does the Fourier Trick Simplify Legendre Polynomial Integrals?

[leonne]

Homework Statement

Hey
I am confussed about how the Fourier trick works. In my book they have an example ...=integral from 0 to pie Pl(cos@)Pl'(cos@)sin(@) d@
then somehow they get ={0 if l' does not equal l and 2/2l+1 if l'=l

Thanks

Homework Equations

The Attempt at a Solution

 

[fzero]

I'm not sure what you mean by "Fourier trick". The orthogonality relationship you are referring to is a result of the Legendre polynomials $$P_n(x)$$ being an orthonormal basis for square-integrable functions on the interval [-1,1]. It's in completely the same spirit as vectors in $$\mathbb{R}^k$$ along with the dot product, but here you have functions as vectors and integrals as your inner product.

It's a bit long to type out a derivation of the orthonormality, but you can find a derivation at http://math.arizona.edu/~zakharov/Legendre Polynomials.pdf

 

[leonne]

o ok thxs ill check it out in the book they say "using the "Fourier trick"

 

[Bhumble]

Fourier's trick works by picking out the Legendre polynomials that contribute to the function of interest by converting them into Dirac Delta functions. I don't remember exactly how it works but I remember that was the gist of it.

 

[fzero]

Fourier's trick works by picking out the Legendre polynomials that contribute to the function of interest by converting them into Dirac Delta functions. I don't remember exactly how it works but I remember that was the gist of it.

Sure if that's the question, then it's the same as writing a general polynomial (square-integrable, etc) in terms of the basis of Legendre polynomials. As in the case of Euclidean vectors, you can compute the coefficients by taking inner products with the basis vectors.