Fourier's Theorem

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

Let $f \left({x}\right)$ be a real function which is defined and bounded on the interval $\left[{\alpha \,.\,.\, \alpha + 2 \pi}\right]$.

Let $f$ satisfy the Dirichlet conditions on $\left[{\alpha \,.\,.\, \alpha + 2 \pi}\right]$:

Outside the interval $\left[{\alpha \,.\,.\, \alpha + 2 \pi}\right]$, let $f$ be periodic and defined such that:
 * $f \left({x}\right) = f \left({x + 2 \pi}\right)$

Let $f$ be defined by the Fourier series:


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

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

Then $(1)$ converges to the sum:
 * $\displaystyle \frac 1 2 \left({\lim_{x \mathop \to 0^+} f \left({x}\right) + \lim_{x \mathop \to 0^-} f \left({x}\right)}\right)$

where the $\lim$ symbols denote the limit from the right and the limit from the left.