Fourier Series for Odd Function over Symmetric Range

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

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


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

where for all $n \in \Z_{> 0}$:
 * $b_n = \displaystyle \frac 2 l \int_0^l f \left({x}\right) \sin \frac {n \pi x} l \rd x$

Proof
By definition of the Fourier series for $f$:


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

From Fourier Cosine Coefficients for Odd Function over Symmetric Range:


 * $a_n = 0$

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

From Fourier Sine Coefficients for Odd Function over Symmetric Range


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

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

Also see

 * Fourier Series for Even Function over Symmetric Range