Definition:Fourier Series/Range 2 Pi

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

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

Let:

Then:


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

is called the Fourier Series for $f$.

Fourier Series on General Range
The range of the Fourier series may be 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 $\openint 0 {2 \pi}$ or $\openint {-\pi} \pi$.

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