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:


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

Then for all $n \in \Z_{\ge 0}$:
 * $b_n = 0$

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


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

By definition of Fourier series:


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

From Sine Function is Odd:
 * $\sin a = -\sin \left({-a}\right)$

for all $a$.

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

The result follows from Definite Integral of Odd Function:
 * $\displaystyle \frac 1 l \int_{-l}^l f \left({x}\right) \sin \frac {n \pi x} l \, \mathrm d x = 0$

Also see

 * Fourier Cosine Coefficients for Even Function over Symmetric Range


 * Fourier Sine Coefficients for Odd Function over Symmetric Range
 * Fourier Cosine Coefficients for Odd Function over Symmetric Range