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:

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.