Fourier Series

What

A way to represent a periodic function as the sum of simple sine and cosine waves.
These sine and cosine functions, also known as harmonics, have frequencies that are integer multiples of a fundamental frequency.

How

Let’s denote the function we want to represent as $f(x)$ and the Fourier series as $F(x)$. If the function $f(x)$ is periodic and integrable over the interval $[-L,L]$, then the Fourier series representation of $f(x)$ is given by:

$F(x)=a_0+\sum _{n=1}^{\infty }[a_n\cos (\frac{n\pi x}{L})+b_n\sin (\frac{n\pi x}{L})]$
where:

• $a_0=\frac{1}{2L}{\int }_{-L}^{L}f(x)dx$,
• $a_n=\frac{1}{L}{\int }_{-L}^{L}f(x)\cos (\frac{n\pi x}{L})dx$,
• $b_n=\frac{1}{L}{\int }_{-L}^{L}f(x)\sin (\frac{n\pi x}{L})dx$.

Links

• Series