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The Fourier series is an example of a trigonometric series. [2] By expressing a function as a sum of sines and cosines, many problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Joseph Fourier to find solutions to the heat equation. This ...
A number of authors, notably Jean le Rond d'Alembert, and Carl Friedrich Gauss used trigonometric series to study the heat equation, [20] but the breakthrough development was the 1807 paper Mémoire sur la propagation de la chaleur dans les corps solides by Joseph Fourier, whose crucial insight was to model all functions by trigonometric series ...
In addition, the improved sensitivity and speed have opened up new areas of application. Spectra can be measured in situations where very little energy reaches the detector. Fourier transform infrared spectroscopy is used in geology, [11] chemistry, materials, botany [12] and biology research fields. [13]
The method of Fourier-transform spectroscopy can also be used for absorption spectroscopy. The primary example is " FTIR Spectroscopy ", a common technique in chemistry. In general, the goal of absorption spectroscopy is to measure how well a sample absorbs or transmits light at each different wavelength.
Spectroscopy is used in physical and analytical chemistry because atoms and molecules have unique spectra. As a result, these spectra can be used to detect, identify and quantify information about the atoms and molecules. Spectroscopy is also used in astronomy and remote sensing on Earth. Most research telescopes have spectrographs.
An Elementary Treatise on Fourier's Series: And Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical Physics (2 ed.). Ginn. p. 30. Carslaw, Horatio Scott (1921). "Chapter 7: Fourier's Series". Introduction to the Theory of Fourier's Series and Integrals, Volume 1 (2 ed.). Macmillan and Company. p. 196.
This includes the study of the notions of Fourier series and Fourier transforms (Fourier analysis), and of their generalizations. Harmonic analysis has applications in areas as diverse as music theory, number theory, representation theory, signal processing, quantum mechanics, tidal analysis, and neuroscience.
Fourier series are an important tool in real analysis. 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 typically occurs and is handled within the branch mathematics ...