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
$$b_n = \frac{2}{L} \int_0^L f(x) \sin \frac{n \pi x}{L} \; dx, \quad n = 1,2, \ldots$$