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Fourier Transform -- from Wolfram MathWorld

The Fourier transform is a generalization of the complex Fourier series in the limit as L->infty. Replace the discrete A_n with the continuous F(k)dk while letting n/L->k. Then change the sum to an integral, and the equations become f(x) = int_(-infty)^inftyF(k)e^(2piikx)dk (1) F(k) = int_(-infty)^inftyf(x)e^(-2piikx)dx. (2) Here, F(k) = F_x[f(x)](k) (3) = int_(-infty)^inftyf(x)e^(-2piikx)dx (4) is called the forward (-i) Fourier transform, and f(x) = F_k^(-1)[F(k)](x) (5) =...



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Fourier Transform -- from Wolfram MathWorld

https://mathworld.wolfram.com/FourierTransform.html

The Fourier transform is a generalization of the complex Fourier series in the limit as L->infty. Replace the discrete A_n with the continuous F(k)dk while letting n/L->k. Then change the sum to an integral, and the equations become f(x) = int_(-infty)^inftyF(k)e^(2piikx)dk (1) F(k) = int_(-infty)^inftyf(x)e^(-2piikx)dx. (2) Here, F(k) = F_x[f(x)](k) (3) = int_(-infty)^inftyf(x)e^(-2piikx)dx (4) is called the forward (-i) Fourier transform, and f(x) = F_k^(-1)[F(k)](x) (5) =...



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https://mathworld.wolfram.com/FourierTransform.html

Fourier Transform -- from Wolfram MathWorld

The Fourier transform is a generalization of the complex Fourier series in the limit as L->infty. Replace the discrete A_n with the continuous F(k)dk while letting n/L->k. Then change the sum to an integral, and the equations become f(x) = int_(-infty)^inftyF(k)e^(2piikx)dk (1) F(k) = int_(-infty)^inftyf(x)e^(-2piikx)dx. (2) Here, F(k) = F_x[f(x)](k) (3) = int_(-infty)^inftyf(x)e^(-2piikx)dx (4) is called the forward (-i) Fourier transform, and f(x) = F_k^(-1)[F(k)](x) (5) =...

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      Fourier Transform -- from Wolfram MathWorld
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      The Fourier transform is a generalization of the complex Fourier series in the limit as L->infty. Replace the discrete A_n with the continuous F(k)dk while letting n/L->k. Then change the sum to an integral, and the equations become f(x) = int_(-infty)^inftyF(k)e^(2piikx)dk (1) F(k) = int_(-infty)^inftyf(x)e^(-2piikx)dx. (2) Here, F(k) = F_x[f(x)](k) (3) = int_(-infty)^inftyf(x)e^(-2piikx)dx (4) is called the forward (-i) Fourier transform, and f(x) = F_k^(-1)[F(k)](x) (5) =...
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      The Fourier transform is a generalization of the complex Fourier series in the limit as L->infty. Replace the discrete A_n with the continuous F(k)dk while letting n/L->k. Then change the sum to an integral, and the equations become f(x) = int_(-infty)^inftyF(k)e^(2piikx)dk (1) F(k) = int_(-infty)^inftyf(x)e^(-2piikx)dx. (2) Here, F(k) = F_x[f(x)](k) (3) = int_(-infty)^inftyf(x)e^(-2piikx)dx (4) is called the forward (-i) Fourier transform, and f(x) = F_k^(-1)[F(k)](x) (5) =...

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      Fourier Transform -- from Wolfram MathWorld
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      The Fourier transform is a generalization of the complex Fourier series in the limit as L->infty. Replace the discrete A_n with the continuous F(k)dk while letting n/L->k. Then change the sum to an integral, and the equations become f(x) = int_(-infty)^inftyF(k)e^(2piikx)dk (1) F(k) = int_(-infty)^inftyf(x)e^(-2piikx)dx. (2) Here, F(k) = F_x[f(x)](k) (3) = int_(-infty)^inftyf(x)e^(-2piikx)dx (4) is called the forward (-i) Fourier transform, and f(x) = F_k^(-1)[F(k)](x) (5) =...

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      Fourier Transform -- from Wolfram MathWorld
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      The Fourier transform is a generalization of the complex Fourier series in the limit as L->infty. Replace the discrete A_n with the continuous F(k)dk while letting n/L->k. Then change the sum to an integral, and the equations become f(x) = int_(-infty)^inftyF(k)e^(2piikx)dk (1) F(k) = int_(-infty)^inftyf(x)e^(-2piikx)dx. (2) Here, F(k) = F_x[f(x)](k) (3) = int_(-infty)^inftyf(x)e^(-2piikx)dx (4) is called the forward (-i) Fourier transform, and f(x) = F_k^(-1)[F(k)](x) (5) =...
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