Derivation of Fourier Series over General Range

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

Let:
 * $\displaystyle f \sim \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty \paren {a_n \cos \frac {n \pi x} l + b_n \sin \frac {n \pi x} l}$

The Fourier coefficients for $f$ are calculated by:

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

Then let:
 * $\map \phi \xi \equiv \map f x$

Thus:

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

Thus $\phi$ is defined and bounded on $\closedint \beta {\beta + 2 \pi}$.

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

where:

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

and so:

and so:


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