Benjamin's Maths World

Integrating a function using Legendre Polynomials and their zeros.

Expanding a function using Legendre Polynomials to produce a series.

Legendre polynomials as the eigenfunctions of the Legendre Equation: Orthogonality, Rodrigues’ Formula, Generating Function.

Constructing Green’s Function for Hermitian Operators as the Superposition of Eigenfunctions.

The method of Green’s Function in the context of Sturm-Liouville boundary value problems.

When the formal term-by-term differentiation and integration of a Fourier series works.

Bessel’s Inequality and Parseval’s Identity: Fourier Series converges to the original function in the \( L^2 \) sense.

Spectral Theorem for Hermitian Operators, leading to the Fourier Basis/Series of the form \( f(x) = \frac{a_0}{2} + a_1 \cos(x) + a_2 \cos(2x) + \cdots + b_1 \sin(x) + b_2 \sin(2x) + \cdots \).

Defining Inner Product Spaces and Hermitian Operators. Properties: Eigenvalues/Eigenfunctions are real/orthogonal.

Definition of Sturm-Liouville Equation \( \mathcal{L}y = \lambda\rho(x) y \) where \( \mathcal{L} = -[\frac{d}{dx}(p(x)\frac{d}{dx}) + q(x)] \).