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

 

    a₀/2 + (a₁cos x + b₁sin x) + (a₂cos 2x + b₂sin 2x) + ...

 

In this series, the nth 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.