Definitions for fourier series
fouri·er se·ries

Princeton's WordNetRate this definition:0.0 / 0 votes

1. Fourier seriesnoun

   the sum of a series of trigonometric expressions; used in the analysis of periodic functions

WiktionaryRate this definition:0.0 / 0 votes

1. Fourier seriesnoun

   a series of cosine and sine functions or complex exponentials resulting from the decomposition of a periodic function

2. Etymology: Named after, a French mathematician

WikipediaRate this definition:0.0 / 0 votes

1. Fourier series

   A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or period), the number of components, and their amplitudes and phase parameters. With appropriate choices, one cycle (or period) of the summation can be made to approximate an arbitrary function in that interval (or the entire function if it too is periodic). The number of components is theoretically infinite, in which case the other parameters can be chosen to cause the series to converge to almost any well behaved periodic function (see Pathological and Dirichlet–Jordan test). The components of a particular function are determined by analysis techniques described in this article. Sometimes the components are known first, and the unknown function is synthesized by a Fourier series. Such is the case of a discrete-time Fourier transform. Convergence of Fourier series means that as more and more components from the series are summed, each successive partial Fourier series sum will better approximate the function, and will equal the function with a potentially infinite number of components. The mathematical proofs for this may be collectively referred to as the Fourier Theorem (see § Convergence). The figures below illustrate some partial Fourier series results for the components of a square wave. Another analysis technique (not covered here), suitable for both periodic and non-periodic functions, is the Fourier transform, which provides a frequency-continuum of component information. But when applied to a periodic function all components have zero amplitude, except at the harmonic frequencies. The inverse Fourier transform is a synthesis process (like the Fourier series), which converts the component information (often known as the frequency domain representation) back into its time domain representation. Since Fourier's time, many different approaches to defining and understanding the concept of Fourier series have been discovered, all of which are consistent with one another, but each of which emphasizes different aspects of the topic. Some of the more powerful and elegant approaches are based on mathematical ideas and tools that were not available in Fourier's time. Fourier originally defined the Fourier series for real-valued functions of real arguments, and used the sine and cosine functions as the basis set for the decomposition. Many other Fourier-related transforms have since been defined, extending his initial idea to many applications and birthing an area of mathematics called Fourier analysis.

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1. fourier series

   A Fourier series is a mathematical tool used in analysis that provides a representation of a periodic function or signal as an infinite sum of sines and cosines. It's typically used in mathematics, physics, and engineering to analyze periodic functions. Named after French mathematician Jean-Baptiste Joseph Fourier, this series has major applications in heat transfer, vibration analysis, digital signal processing, image analysis, and much more. It offers a basis for Decomposition of functions or signals into their constituent sinusoidal components.

WikidataRate this definition:0.0 / 0 votes

1. Fourier series

   In mathematics, a Fourier series decomposes periodic functions or periodic signals into the sum of a set of simple oscillating functions, namely sines and cosines. The study of Fourier series is a branch of Fourier analysis. The Fourier series is named in honour of Jean-Baptiste Joseph Fourier, 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, 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 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.

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Numerology

1. Chaldean Numerology

   The numerical value of fourier series in Chaldean Numerology is: 5

2. Pythagorean Numerology

   The numerical value of fourier series in Pythagorean Numerology is: 5

References

1. ^ Princeton's WordNet
   http://wordnetweb.princeton.edu/perl/webwn?s=fourier series
2. ^ Wiktionary
   https://en.wiktionary.org/wiki/Fourier_Series
3. ^ Wikipedia
   https://en.wikipedia.org/wiki/Fourier_Series
4. ^ ChatGPT
   https://chat.openai.com
5. ^ Wikidata
   https://www.wikidata.org/w/index.php?search=fourier series