“Fourier series”的概念、定义、翻译、参考文献-科学参考

Fourier series

Physics

An expansion of a periodic function as a series of trigonometric functions. Thus,

$f (x) = a_{0} + (a_{1} cos x + b_{1} sin x) + (a_{2} cos 2 x + b_{2} sin 2 x) + \dots,$

where a₀, a₁, b₁, b₂, etc., are constants, called Fourier coefficients. The series is used in Fourier analysis.

Mathematics

The infinite series $\frac{1}{2} a_{0} + \sum_{n = 1}^{\infty} (a_{n} cos (n π x / L) + b_{n} sin (n π x / L))$ , where a_n and b_n are the Fourier coefficients. The Fourier series defines a function of period 2L; the convergence of the series and the magnitude of the Fourier coefficients are highly related to the order of differentiability of the function f. Fourier series are used to decompose a waveform into component waves of different frequencies and amplitudes, allowing identification of different sources from background or random noise in a signal. Fourier series are also widely used in the solution of partial differential equations.

Statistics

If f is an integrable function on (−π, π) then the Fourier series for f is $Fourier series$ where $Fourier series$

Chemistry

An expansion of a periodic function as a series of trigonometric functions. Thus,
$f (x) = a_{0} + (a_{1} cos x + b_{1} sin x) + (a_{2} cos 2 x + b_{2} sin 2 x) + …,$
where a₀, a₁, b₁, b₂, etc., are constants, called Fourier coefficients. The series was first formulated by Joseph Fourier and is used in Fourier analysis.

Computer

The infinite trigonometric series
$\frac{1}{2} a_{0} + \sum_{n = 1}^{\infty} (a_{n} cos n x + b_{n} sin n x)$
By suitable choice of the coefficients a_i and b_i, the series can be made equal to any function of x defined on the interval (− π‎, π‎). If f is such a function, the Fourier coefficients are given by the formulas
$\begin{array}{l} a_{n} = (1 / π) \int_{- π}^{π} f (x) cos n x d x \\ (n = 0, 1, 2, \dots) \\ b_{n} = (1 / π) \int_{- π}^{π} f (x) sin n x d x \\ (n = 1, 2, \dots) \end{array}$

单词	Fourier series
释义	Fourier series Physics An expansion of a periodic function as a series of trigonometric functions. Thus, $f (x) = a_{0} + (a_{1} cos x + b_{1} sin x) + (a_{2} cos 2 x + b_{2} sin 2 x) + \dots,$ where a₀, a₁, b₁, b₂, etc., are constants, called Fourier coefficients. The series is used in Fourier analysis. Mathematics The infinite series $\frac{1}{2} a_{0} + \sum_{n = 1}^{\infty} (a_{n} cos (n π x / L) + b_{n} sin (n π x / L))$ , where a_n and b_n are the Fourier coefficients. The Fourier series defines a function of period 2L; the convergence of the series and the magnitude of the Fourier coefficients are highly related to the order of differentiability of the function f. Fourier series are used to decompose a waveform into component waves of different frequencies and amplitudes, allowing identification of different sources from background or random noise in a signal. Fourier series are also widely used in the solution of partial differential equations. Statistics If f is an integrable function on (−π, π) then the Fourier series for f is $Fourier series$ where $Fourier series$ Chemistry An expansion of a periodic function as a series of trigonometric functions. Thus, $f (x) = a_{0} + (a_{1} cos x + b_{1} sin x) + (a_{2} cos 2 x + b_{2} sin 2 x) + …,$ where a₀, a₁, b₁, b₂, etc., are constants, called Fourier coefficients. The series was first formulated by Joseph Fourier and is used in Fourier analysis. Computer The infinite trigonometric series $\frac{1}{2} a_{0} + \sum_{n = 1}^{\infty} (a_{n} cos n x + b_{n} sin n x)$ By suitable choice of the coefficients a_i and b_i, the series can be made equal to any function of x defined on the interval (− π‎, π‎). If f is such a function, the Fourier coefficients are given by the formulas $\begin{array}{l} a_{n} = (1 / π) \int_{- π}^{π} f (x) cos n x d x \\ (n = 0, 1, 2, \dots) \\ b_{n} = (1 / π) \int_{- π}^{π} f (x) sin n x d x \\ (n = 1, 2, \dots) \end{array}$
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