Definition:Fourier Series/Formulation 1

Definition
Let $\alpha, l \in \R$ be real numbers.

Let $f: \R \to \R$ be a function such that $\displaystyle \int_\alpha^{\alpha + 2 l} f \left({x}\right) \, \mathrm d x$ converges absolutely.

Let:

Then:


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

is the Fourier Series for $f$.

Also see

 * Derivation of Fourier Series over General Range, which provides the justification for this definition