10.4 | Fourier Cosine and Sine Series


    Let \(f(x)\) be a piecewise function continuous on \([0,L].\) The Fourier cosine series of \(f(x)\) on \([0,L]\) is
    $$
    \frac{a_0}{2} + \sum_{n=1}^\infty a_n \cos \frac{n \pi x}{L},
    $$
    where
    $$
    a_n = \frac{2}{L} \int_0^L f(x) \cos \frac{n \pi x}{L} \; dx, \quad n = 0,1, \ldots
    $$
    The Fourier sine series of \(f(x)\) on \([0,L]\) is
    $$
    \sum_{n=1}^\infty b_n \sin \frac{n \pi x}{L},
    $$
    where
    $$
    b_n = \frac{2}{L} \int_0^L f(x) \sin \frac{n \pi x}{L} \; dx, \quad n = 1,2, \ldots
    $$

    E 10.4 Exercises