Fourier Series

A Fourier Series is a mathematical way to represent a periodic function or signal as a sum of sinusoids (SINE & COSINE WAVES).

History

The Fourier Series was named after Jean-Baptiste Joseph Fourier, and was aided by previous investigations by Leonhard Euler and Daniel Bernoulli. Fourier made the series to solve the heat equation of a metal plate. He published his results in 1807. In fact the heat equation of a simple heat system resembled that of a simple sinusoid,  so the more complicated systems could be acheived as a linear combination of a series of sinusoids, thus the Fourier series. The heat equation is a partial differential equation. Other similar applications included the orbit of celestial bodies.
However Fourier's finding were informal, so they were later formalized with Peter Dirchlet and Bernhard Riemann.

Definition

\(s\left(x\right)\) represents a function of real variable \(x\), that is integrable on the interval \(\left[x_0\ ,\ x_0+P\right]\) for real numbers \(x_0\) and \(P\).
\(s_N\left(x\right)=\frac{A_0}{2}+\sum_{n=1}^NA_n\cdot\sin\left(\frac{2\pi nx}{P}+\phi_n\right),\ for\ integer\ N\ \ge1.\)