# Fourier series

A Fourier series, named after Joseph Fourier, is the expansion
of a periodic function as an infinite sum
of sines and cosines of various frequencies and amplitudes. This is similar to the approximation
of an irrational number by a sum of a series of rational numbers (or a decimal
expansion). Most bounded, periodic functions (on the domain -π ≤ *x* ≤ π) can be analyzed as a sum of simple harmonic components.
Thus if *f*(*x*) is such a function, with *x* taking
values between -π and π, so that

*f*(*x* + 2π) = *f*(*x*)

it may be expressed as

*a*_{0}/2 + (*a*_{1}cos *x* + *b*_{1}sin *x*) + (*a*_{2}cos 2*x* + *b*_{2}sin 2*x*) + ...

In this series, the *n*th coefficients, *a*_{n} and *b*_{n}, are given by

*a*_{n} = 1/π(∫(from -π to π) *f*(*x*)cos(*nx*) *dx*), and

*b*_{n} = 1/π(∫(from -π to π) *f*(*x*)sin(*nx*) *dx*)

Human ears effectively produce Fourier series automatically from complex sounds. Tiny hairs, known as cilia, vibrate at different specific frequencies. When a wave enters the ear, the cilia vibrate if the wave function contains any component of the corresponding frequency. This enables the hearer to distinguish sounds of various pitches.

Fourier series are used a great deal in science and engineering to find solutions to partial differential equations, such as those in problems involving heat flow. They can also be used to construct some pathological functions such as ones that are continuous but nowhere differentiable. The study and computation of Fourier series is known as harmonic analysis.