# Fourier series

Fourier coefficientFourier expansionFourier coefficientsFourier modesFourierFourier decompositioncosine seriesFourier theoremFourier's theoremHilbert space
In mathematics, a Fourier series is a periodic function composed of harmonically related sinusoids, combined by a weighted summation.wikipedia
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### Sturm–Liouville theory

Sturm–Liouville problemSturm–LiouvilleSturm–Liouville operator
. We know that the solutions of a S–L problem form an orthogonal basis, and we know from Fourier series that this set of sinusoidal functions is an orthogonal basis.

### Multidimensional transform

multidimensional Fourier transform
Discrete Fourier transforms are often used to solve partial differential equations, where again the DFT is used as an approximation for the Fourier series (which is recovered in the limit of infinite N).

### Mathematics

mathematicalmathmathematician
In mathematics, a Fourier series is a periodic function composed of harmonically related sinusoids, combined by a weighted summation.

### Sine wave

sinusoidalsinusoidsine
In mathematics, a Fourier series is a periodic function composed of harmonically related sinusoids, combined by a weighted summation.

### Daniel Bernoulli

BernoulliDanielBernoulli, Daniel
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.

### Heat equation

heat diffusionheatanalytic theory of heat
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 (Analytical theory of heat) in 1822.

The first announcement of this great discovery was made by Fourier in 1807, before the French Academy.

### Eigenvalues and eigenvectors

eigenvalueeigenvalueseigenvector
These simple solutions are now sometimes called eigensolutions.

### Linear combination

linear combinationslinearly combined(finite) left ''R''-linear combinations
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.

### Superposition principle

superpositionlinear superpositionsuperpose
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.

### Function (mathematics)

functionfunctionsmathematical function
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.

### Integral

integrationintegral calculusdefinite integral
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.

### Electrical engineering

electrical engineerelectricalElectrical and Electronics Engineering
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.

### Oscillation

oscillatorvibrationoscillators
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.

### Acoustics

acousticacousticianacoustical
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.

### Optics

opticalopticoptical system
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.

### Signal processing

signal analysissignalsignal processor
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.

### Digital image processing

image processingimageprocessing
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.

### Quantum mechanics

quantum physicsquantum mechanicalquantum theory
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.

### Econometrics

econometriceconometricianeconometric analysis
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.

### Thin-shell structure

shellsshellthin-shell
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.

### Euler's formula

complex exponentialcomplex exponentialsEuler's exponential formula
The customary form for generalizing to complex-valued s(x) (next section) is obtained using Euler's formula to split the cosine function into complex exponentials.

### Complex conjugate

complex conjugationconjugateconjugation
Here, complex conjugation is denoted by an asterisk:

### Hertz

MHzkHzHz
When variable x has units of seconds, f has units of hertz.