Fourier Series

• e.g. plotting a Square Wave as a sum of sinusoids
• Any function can be composed of a sum of Even Functions and Odd Functions
  • Fourier Cosine Series
  • Fourier Sine Series

$$f(t)=\mathbb{E}(t)+\mathbb{O}(t)$$ $$f(t)=\frac{1}{\pi }\sum _{m=0}^{\infty }{\mathbb{E}}_m\cos (mt)+\frac{1}{\pi }\sum _{m=0}^{\infty }{\mathbb{O}}_m\sin (mt)$$ $$\text{where }{\mathbb{E}}_m=\int \mathbb{E}(t)\cos (mt)dt=\int f(t)\cos (mt)dt$$ $$\text{where }{\mathbb{O}}_m=\int \mathbb{O}(t)\sin (mt)dt=\int f(t)\sin (mt)dt$$

The $f_{\mathbb{E}}$ in the odd integral and the $f_{\mathbb{O}}$ in the even integral both integrate to zero, which is why we just simplify to $f(t)$