A(x) and B(x) are orthogonal on (a,b) if the integral (from a to b) of A(x) B(x) dx gives 0.
If A and/or B are complex, then you must complex conjugate one of the two functions in the definition. (Either one, it's equivalent)
Definition of an orthogonal set of functions - any pair from the set will be orthogonal.
Examples (sin(nx), Exp[inx], and now, Legendre polynomials)
Discussion of complete set
Proof that Legendre poly's are orthogonal
Definition of the norm of a function, and discussion of "orthonormal set".
Example of "Fourier-Legendre" expansion of a function. Derived general formula, then did specific example.
Many old intuitions/rules from Fourier series hold, e.g. Dirichlet.
Discussion of value of Legendre functions (problems with spherical symmetry, also provides best-fit [least squares] polynomial to a function).
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