What is a Fourier series?

A Fourier series is a way to represent a function as a combination of sine and cosine waves. It is named after the French mathematician and physicist Joseph Fourier, who first introduced the concept in the early 19th century. The idea behind a Fourier series is that any periodic function can be approximated by an infinite sum of sine and cosine functions with different frequencies and amplitudes.

Representation

The general form of a Fourier series for a periodic function f(x) with period 2π is given by:

f(x) = a₀ + Σ(a_ncos(nx) + b_nsin(nx))

where n ranges from 1 to infinity, and a₀, a_n, and b_n are the coefficients that determine the amplitude of the cosine and sine terms.

Applications

Fourier series are widely used in various fields such as signal processing, image analysis, and physics. They are used to analyze and process periodic signals, decompose functions into simpler components, and solve partial differential equations. The Fourier series has paved the way for more advanced concepts like Fourier transforms and frequency analysis, making it a fundamental tool in mathematics and engineering.