Fourier Sine Coefficients for Odd Function over Symmetric Range

Theorem
Let $\map f x$ be an odd real function defined on the interval $\closedint {-l} l$.

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


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

Then for all $n \in \Z_{> 0}$:
 * $b_n = \displaystyle \frac 2 l \int_0^l \map f x \sin \frac {n \pi x} l \rd x$

Proof
As suggested, let the Fourier series of $\map f x$ be expressed as:


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

By definition of Fourier series:


 * $b_n = \displaystyle \frac 1 l \int_{-l}^{-l + 2 l} \map f x \sin \frac {n \pi x} l \rd x$

From Sine Function is Odd:
 * $\sin a = -\map \sin {-a}$

for all $a$.

By Odd Function Times Odd Function is Even, $\map f x \sin \dfrac {n \pi x} l$ is even.

Thus:

Also see

 * Fourier Cosine Coefficients for Odd Function over Symmetric Range


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