Definition:Fourier Series/Formulation 2

Definition
Let $a, b \in \R$ be real numbers.

Let $f: \R \to \R$ be a function such that $\displaystyle \int_a^b f \left({x}\right) \, \mathrm d x$ converges absolutely.

Let:

Then:


 * $\displaystyle \frac {A_0} 2 + \sum_{m \mathop = 1}^\infty \left({A_m \cos \frac {2 m \pi \left({x - a}\right)} {b - a} + B_m \sin \frac {2 m \pi \left({x - a}\right)} {b - a} }\right)$

is the Fourier Series for $f$.

Also defined as
The form given here is more general than that usually given.

The usual form is one of the cases where $a = 0$ and $b = 2 l$, or $a = -l$ and $b = l$, thus giving a range of integration of either $\left[{0 \,.\,.\, 2 l}\right]$ or $\left[{-l \,.\,.\, l}\right]$.

The actual range may often be chosen for convenience of analysis.

Also see

 * Equivalence of Definitions of General Fourier Series


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