Definition:Fourier Series

Definition
Let $f:\R \to \R$ be a function such that $\displaystyle \int_{-\pi}^\pi f(t)\,\mathrm{d}t$ converges absolutely. If we set


 * $\displaystyle \pi a_n = \int_{-\pi}^\pi f(t) \cos(nt)\,\mathrm{d}t, \qquad \pi b_n=\int_{-\pi}^\pi f(t)\sin(nt)\,\mathrm{d}t$,

then


 * $\displaystyle \frac{a_0}{2} + \sum\limits_{n\mathop=1}^\infty \left({ a_n \cos(nx) + b_n \sin(nx) }\right)$

is called the Fourier Series for $f$.