more on Fourier Series.

## decomposition of functions to even and odd

Suppose we have any function with period \(L\) over \([-\frac{L}{2}, \frac{L}{2}]\), we can write this as a sum of even and odd functions:

\begin{equation} f(x) = \frac{1}{2} (f(x) - f(-x)) + \frac{1}{2} (f(x) + f(-x)) \end{equation}

And because of this fact, we can actually take each part and break it down individually as a Fourier Series because sin and cos are even and odd parts.

So we can take the first part, which is odd, and break it down using \(a_{n} \sin (k\omega x)\).

We can take the second part, which is odd, and break it down using \(b_{n} \cos (k\omega x)\).

If you then assume periodicity over the interval you care about \(L\), suddenly you can decompose it to a Fourier Series.