Fourier Sine Series S.L problem: [{\begin{matrix}\frac{d^{2}y}{dx^{2}}+\lambda y=0\ y(0)=0=y(l)\end{matrix}] [0\le x\le l] p = 1, q = 1 proper [y_{n}(x)=B_{n}\sin(\frac{n\pi x}{l})] [\lambda_{n}=(\frac{n\pi}{l})^{2}] [n=1,2,3,...] [\int_{0}^{l}y_{n}(x)y_{m}(x)dx = \delta_{nm}] [\delta_{nm} = \begin{cases} 1 & \text{if } n = m \ 0 & \text{if } n \neq m \end{cases}] [\int_{0}^{l}\sin(nx)\sin(mx)dx = \frac{l}{2}\delta_{nm}] More generally: If f(x) is continuous at x, then for all x: [f(x) = \frac{1}{2}[f(x^{+}) + f(x^{-})]] If f(x) is discontinuous at x₀, then: [f(x) = \frac{1}{2}[f(x_{0}^{+}) + f(x_{0}^{-})]] Fourier Cosine Series Motivation: S.L problem: [y''(x) + \lambda y = 0] [y'(0) = 0 = y'(l)] [0 \leq x \leq l] (homogeneous BC) Solutions: [y_{n}(x) = A_{n}\cos(\frac{n\pi x}{l})] [\lambda_{n} = (\frac{n\pi}{l})^{2}] [n = 0, 1, 2, 3, ...] Given any admissible f(x): [f(x) = \sum_{n=0}^{\infty}A_{n}\cos(\frac{n\pi x}{l})] To find A₀: [A_{0} = \frac{1}{l}\int_{0}^{l}f(x)dx] To find Aₙ: [A_{n} = \frac{2}{l}\int_{0}^{l}f(x)\cos(\frac{n\pi x}{l})dx] [n = 1, 2, 3, ...]