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:

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 $f$.

Fourier Series on General Range
The range of the fourier series may be further extended to any general real interval.

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.

Also see

 * Coefficients of Cosine Terms in Convergent Trigonometric Series
 * Coefficients of Sine Terms in Convergent Trigonometric Series