Fourier Sine Series

Because $\sin (mt)$ is an Odd Function we can write any odd function as

$$\mathbb{O}(t)=\frac{1}{\pi }\sum _{m=0}^{\infty }{\mathbb{O}}_m\sin (mt)$$

where the set of coefficients of ${\mathbb{O}}_m$ define the series and $\mathbb{O}(t)$ is over $(-\pi ,\pi )$ This function only sometimes works, but we should assume it will work generally in Physics

Finding the coefficients of $\mathbb{O}(t)$

Exactly the same as the derivation in Fourier Cosine Series, except with sines.

$${\mathbb{O}}_m={\int }_{-\pi }^{\pi }\mathbb{O}(t)\sin (mt)dt$$