# Fourier Series

In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a (possibly infinite) set of simple oscillating functions, namely sines and cosines (or complex exponentials). The study of Fourier series is a branch of Fourier analysis.

The Fourier series is named in honour of Jean-Baptiste Joseph Fourier (1768–1830), who made important contributions to the study of trigonometric series, after preliminary investigations by Leonhard Euler, Jean le Rond d'Alembert, and Daniel Bernoulli. Fourier introduced the series for the purpose of solving the heat equation in a metal plate, publishing his initial results in his 1807 Mémoire sur la propagation de la chaleur dans les corps solides (Treatise on the propagation of heat in solid bodies), and publishing his Théorie analytique de la chaleur in 1822. Early ideas of decomposing a periodic function into the sum of simple oscillating functions date back to the 3rd century BC, when ancient astronomers proposed an empiric model of planetary motions, based on deferents and epicycles.

The heat equation is a partial differential equation. Prior to Fourier's work, no solution to the heat equation was known in the general case, although particular solutions were known if the heat source behaved in a simple way, in particular, if the heat source was a sine or cosine wave. These simple solutions are now sometimes called eigensolutions. Fourier's idea was to model a complicated heat source as a superposition (or linear combination) of simple sine and cosine waves, and to write the solution as a superposition of the corresponding eigensolutions. This superposition or linear combination is called the Fourier series.

From a modern point of view, Fourier's results are somewhat informal, due 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.

Although the original motivation was to solve the heat equation, it later became obvious that the same techniques could be applied to a wide array of mathematical and physical problems, and especially those involving linear differential equations with constant coefficients, for which the eigensolutions are sinusoids. The Fourier series has many such applications in electrical engineering, vibration analysis, acoustics, optics, signal processing, image processing, quantum mechanics, econometrics, thin-walled shell theory, etc.

Read more about Fourier Series:  Revolutionary Article, Birth of Harmonic Analysis, Definition, Properties, Approximation and Convergence of Fourier Series

### Other articles related to "fourier, fourier series, series":

results">Hilbert Space - Applications - Fourier Analysis
... One of the basic goals of Fourier analysis is to decompose a function into a (possibly infinite) linear combination of given basis functions the associated Fourier series ... The classical Fourier series associated to a function f defined on the interval is a series of the form where The example of adding up the first few terms in a ... A significant problem in classical Fourier series asks in what sense the Fourier series converges, if at all, to the function f ...
Approximation and Convergence of Fourier Series - Divergence
... Since Fourier series have such good convergence properties, many are often surprised by some of the negative results ... For example, the Fourier series of a continuous T-periodic function need not converge pointwise ... In 1922, Andrey Kolmogorov published an article entitled "Une série de Fourier-Lebesgue divergente presque partout" in which he gave an example of a Lebesgue-integrable function ...
Half Range Fourier Series
... A half range Fourier series is a Fourier series defined on an interval instead of the more common, with the implication that the analyzed function should be extended to as ... This allows the expansion of the function in a series solely of sines (odd) or cosines (even) ... Calculate the half range Fourier sine series for the function where ...
Spinodal Decomposition - Fourier Transform
... The motivation for the Fourier transform comes from the study of a Fourier series ... In the study of a Fourier series, complicated periodic functions are written as the sum of simple waves mathematically represented by sines and cosines ... that e2πiθ = cos 2πθ + i sin 2πθ, to write Fourier series in terms of the basic waves e2πiθ, with the distinct advantage of simplifying many unwieldy formulas ...
Conjugate Fourier Series
... In the mathematical field of Fourier analysis, the conjugate Fourier series arises by realizing the Fourier series formally as the boundary values of the real part of a holomorphic function on the unit disc ... The imaginary part of that function then defines the conjugate series ... questions of convergence of this series, and its relationship with the Hilbert transform ...

### Famous quotes containing the word series:

I look on trade and every mechanical craft as education also. But let me discriminate what is precious herein. There is in each of these works an act of invention, an intellectual step, or short series of steps taken; that act or step is the spiritual act; all the rest is mere repetition of the same a thousand times.
Ralph Waldo Emerson (1803–1882)