Fourier Cosine Series

Because $\cos (mt)$ is an Even Function we can write any even function as

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

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

Finding the Coefficients

$$\mathbb{E}(t)=\frac{1}{\pi }\sum _{m^{\prime }=0}^{\infty }{\mathbb{E}}_{m^{\prime }}\cos (m^{\prime }t)$$ $${\int }_{-\pi }^{\pi }\mathbb{E}(t)\cos (mt)dt=\frac{1}{\pi }\sum _{m^{\prime }=0}^{\infty }{\int }_{-\pi }^{\pi }{\mathbb{E}}_{m^{\prime }}\cos (m^{\prime }t)\cos (mt)dt$$

Using Kronecker Delta Function, simplify the integral

$${\int }_{-\pi }^{\pi }\mathbb{E}(t)\cos (mt)dt=\frac{1}{\pi }\sum _{m^{\prime }=0}^{\infty }{\mathbb{E}}_{m^{\prime }}\pi {\delta }_{m^{\prime },m}$$

The sum only contributes once when $m^{\prime }=m$, giving one $\pi$

$${\int }_{-\pi }^{\pi }\mathbb{E}(t)\cos (mt)dt=\frac{1}{\pi }{\mathbb{E}}_m\pi$$ $$⟹{\mathbb{E}}_m={\int }_{-\pi }^{\pi }\mathbb{E}(t)\cos (mt)dt$$