\begin{definition}[Definition:Fourier Series/Range 2 Pi]
Let $\alpha \in \R$ be a real number.
Let $f: \R \to \R$ be a function such that $\ds \int_\alpha^{\alpha + 2 \pi} \map f x \rd x$ converges absolutely.
Let:
{{begin-eqn}}
{{eqn | l = a_n
      | r = \dfrac 1 \pi \int_\alpha^{\alpha + 2 \pi} \map f x \cos n x \rd x
}}
{{eqn | l = b_n
      | r = \dfrac 1 \pi \int_\alpha^{\alpha + 2 \pi} \map f x \sin n x \rd x
}}
{{end-eqn}}
Then:
:$\dfrac {a_0} 2 + \ds \sum_{n \mathop = 1}^\infty \paren {a_n \cos n x + b_n \sin n x}$
is called the '''Fourier Series''' for $f$.
\end{definition}