Derivation of Fourier Series over General Range

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

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

The Fourier coefficients for $f$ are calculated by:

Proof
Let $\xi = \dfrac {\pi x} l$.

Then let:
 * $\phi \left({\xi}\right) \equiv f \left({x}\right)$

Thus:

Setting $\beta = \dfrac{\pi \alpha} l$, this allows us:

Thus $\phi$ is defined and bounded on $\left[{\beta \,.\,.\, \beta + 2 \pi}\right]$.

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

where:

We have that:
 * $\dfrac {\mathrm d \xi} {\mathrm d x} = \dfrac \pi l$

and so:

and so:


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