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.

Let:


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

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

is called the Fourier Series for $f$.