Fourier series

Fourier series illustration

Correlation function

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sawtooth pi

Periodic identity function

Fourier heat in a plate

File:Complex Fourier series tracing and stereo playing the letter 'e'.webm Fourier series are a way to represent a function as the sum of simple sine waves. Mathematically, they decompose any periodic function or periodic signal into the sum of a (possibly infinite) set of simple oscillating functions, namely sines and cosines (or, equivalently, complex exponentials). The study of Fourier series is a branch of Fourier analysis named after Joseph Fourier, who showed that representing a function as a sum of trigonometric functions greatly simplifies the study of heat transfer.

Definition

Given a periodic function f(x) with period T, the Fourier series of f is given by:

\[f(x) = a_0 + \sum_{n=1}^{\infty} (a_n \cos(n \omega x) + b_n \sin(n \omega x))\]

where

• n is an integer,
• \(\omega = \frac{2\pi}{T}\) is the angular frequency,
• \(a_0\) is the average of the function over one period,
• \(a_n\) and \(b_n\) are the coefficients for the cosine and sine components, respectively, and are calculated as follows:

\[a_n = \frac{2}{T} \int_{0}^{T} f(x) \cos(n \omega x) \, dx\]

\[b_n = \frac{2}{T} \int_{0}^{T} f(x) \sin(n \omega x) \, dx\]

Applications

Fourier series have a wide range of applications in electrical engineering, signal processing, and acoustics, where they are used to analyze vibration, waves, and other periodic functions. In mathematics, they are used in solving partial differential equations. In quantum mechanics, Fourier series are used to describe the states of a quantum system within a finite interval.

Convergence

The question of convergence of a Fourier series is complex and depends on the properties of the function being represented. Under certain conditions, such as if the function is piecewise continuous and has a finite number of discontinuities, the Fourier series converges to the function at most points. The Dirichlet's theorem on Fourier series gives more precise conditions under which convergence occurs.

Extensions and Generalizations

The concept of Fourier series can be extended to functions of more than one variable, leading to Fourier series in two dimensions and further, to Fourier transforms for non-periodic functions. Another generalization is the Laplace transform, which can be seen as a Fourier transform with complex frequency.

See Also

• Fourier transform
• Laplace transform
• Wave equation
• Heat equation