Fourier Sine Coefficients for Even Function over Symmetric Range

Theorem

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

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

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

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

$b_n = 0$

Proof

As suggested, let the Fourier series of $f \left({x}\right)$ be expressed as:

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

By definition of Fourier series:

 $\displaystyle b_n$ $=$ $\displaystyle \frac 1 l \int_{-l}^{-l + 2 l} f \left({x}\right) \sin \frac {n \pi x} l \, \mathrm d x$ $\displaystyle$ $=$ $\displaystyle \frac 1 l \int_{-l}^l f \left({x}\right) \sin \frac {n \pi x} l \, \mathrm d x$

From Sine Function is Odd:

$\sin a = -\sin \left({-a}\right)$

for all $a$.

By Odd Function Times Even Function is Odd, $f \left({x}\right) \sin \dfrac {n \pi x} l$ is odd.

The result follows from Definite Integral of Odd Function:

$\displaystyle \frac 1 l \int_{-l}^l f \left({x}\right) \sin \frac {n \pi x} l \, \mathrm d x = 0$

$\blacksquare$