Fourier series

The Fourier series, named in honor of Joseph Fourier (1768-1830), is an extremely useful mathematical tool. Intuitively, one can use Fourier series to divide certain large problems into more manageable pieces.

Related Topics:
Joseph Fourier - 1768 - 1830 - Mathematical

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More precisely, a Fourier series, is a representation of a periodic function with period 2π as a sum of periodic functions of the form

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:xmapsto e^{inx},

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which are the harmonics of ei x. By Euler's formula, the series can be expressed equivalently in terms of sine and cosine functions. This can be generalized to periodic functions of any positive period.

Related Topics:
Harmonic - Euler's formula - Sine - Cosine

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Fourier was the first to study systematically such infinite series, after preliminary investigations by Euler, d'Alembert, and Daniel Bernoulli. He applied these series to the solution of the heat equation, publishing his initial results in 1807 and 1811, and publishing his Théorie analytique de la chaleur in 1822. From a modern point of view, Fourier's results are somewhat informal, due in no small part to the lack of a precise notion of function and integral in the early nineteenth century. Later, Dirichlet and Riemann expressed Fourier's results with greater precision and formality.

Related Topics:
Infinite series - Euler - D'Alembert - Daniel Bernoulli - Heat equation - 1807 - 1811 - 1822 - Function - Integral - Dirichlet - Riemann

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Many other Fourier-related transforms have since been defined, extending to other applications the initial idea of representing any periodic function as a superposition of harmonics. This general area of inquiry is now sometimes called harmonic analysis.

Related Topics:
Fourier-related transforms - Superposition - Harmonic analysis

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