# Fourier Series for Even Function over Symmetric Range

## Theorem

Let $f \left({x}\right)$ be an even real function defined on the interval $\left[{-l \,.\,.\, l}\right]$.

Then the Fourier series of $f \left({x}\right)$ can be expressed as:

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

where for all $n \in \Z_{\ge 0}$:

$a_n = \displaystyle \frac 2 l \int_0^l f \left({x}\right) \cos \frac {n \pi x} l \rd x$

## Proof

By definition of the Fourier series for $f$:

$\displaystyle f \left({x}\right) \sim \frac {a_0} 2 + \sum_{n \mathop = 1}^\infty \left({a_n \cos \frac {n \pi x} l + b_n \sin \frac {n \pi x} l}\right)$
$a_n = \displaystyle \frac 2 l \int_0^l f \left({x}\right) \cos \frac {n \pi x} l \rd x$

for all $n \in \Z_{\ge 0}$.

$b_n = 0$

for all $n \in \Z_{\ge 0}$:

$\blacksquare$