Definition:Fourier Series

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

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

Let:


 * $\displaystyle a_n = \dfrac 1 \pi \int_\alpha^{\alpha + 2 \pi} f \left({x}\right) \cos n x \, \mathrm d x$
 * $\displaystyle b_n = \dfrac 1 \pi \int_\alpha^{\alpha + 2 \pi} f \left({x}\right) \sin n x \, \mathrm d x$

Then:


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

is called 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 $\alpha = 0$ or $\alpha = -\pi$, thus giving a range of integration of either $\left[{0 \,.\,.\, 2 \pi}\right]$ or $\left[{-\pi \,.\,.\, \pi}\right]$.

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