# Fourier Series for Odd Function over Symmetric Range

## Theorem

Let $\map f x$ be an odd real function defined on the interval $\openint {-\lambda} \lambda$.

Then the Fourier series of $\map f x$ can be expressed as:

$\map f x \sim \displaystyle \sum_{n \mathop = 1}^\infty b_n \sin \frac {n \pi x} \lambda$

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

$b_n = \displaystyle \frac 2 \lambda \int_0^\lambda \map f x \sin \frac {n \pi x} \lambda \rd x$

## Proof

By definition of the Fourier series for $f$:

$\map f x \sim \dfrac {a_0} 2 + \displaystyle \sum_{n \mathop = 1}^\infty \paren {a_n \cos \frac {n \pi x} \lambda + b_n \sin \frac {n \pi x} \lambda}$
$a_n = 0$

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

$b_n = \displaystyle \frac 2 \lambda \int_0^\lambda \map f x \sin \frac {n \pi x} \lambda \rd x$

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

$\blacksquare$