Fourier analysis

Introduction
Fourier analysis is a method of analysing functions. These functions may be electrical signals (say, from an electronic circuit being tested), pure mathematical functions, or any kind of data being analysed on a computer. Regardless, if the function is single-valued, Fourier analysis can be used to produce an imperfect approximation.

The trigonometric form
Fourier analysis works by breaking down the function being considered into a Fourier Series. The Fourier Series, in simplest terms, is a summation of sine and cosine functions. Each of these trigonometric functions looks something like this:

$$a_n \cos(\omega_n t) + b_n \sin(\omega_n t)$$

The exponential form
Consider f(x) as real valued function $$f(x)=\sum_{n=-\infty}^{\infty} A_n e^{inx}$$