Definition:Fourier Series

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


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

then


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

is called the Fourier Series for $f$.