Definition:Fourier Series

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


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

then


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

is called the Fourier Series for $f$.