Fourier Series and Transform

A Fourier series is an expansion of trigonometric functions to model periodic functions. This method proves useful in the study of harmonic systems as the analysis in a more familiar domain is much simpler than in its original domain. It has a variety of applications ranging from signal processing to quantum mechanics. The Fourier Series is defined as:
[math]\displaystyle{ f(x)=\sum_{n=1}^{\infty}{a_n\cos{(\frac{nx}{L}})}+\sum_{n=1}^{\infty}{b_n\sin{(\frac{nx}{L}})} }[/math]

Intuition

Many physical systems can be modeled by square waves. Consider systems with on-off behavior, similar to an on-and-off switch. A square wave looks like this:
Error creating thumbnail: sh: /usr/bin/convert: No such file or directory Error code: 127