Definition:Fourier Series/Formulation 1

Definition
Let $\alpha \in \R$ be a real number.

Let $\lambda \in \R_{>0}$ be a strictly positive real number.

Let $\openint \alpha {\alpha + 2 \lambda}$ be an open interval of $\R$.

Let $f: \R \to \R$ be a function such that $\displaystyle \int_\alpha^{\alpha + 2 \lambda} \map f x \rd x$ converges absolutely.

Let:

Then:


 * $\displaystyle \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty \paren {a_n \cos \frac {n \pi x} \lambda + b_n \sin \frac {n \pi x} \lambda}$

is the Fourier Series for $f$.

Also see

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