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Inverse Fourier Transform

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The inverse Fourier transform is the integral transform which recovers a function from its Fourier transform. In the convention used in this work (Bracewell 1999, pp. 6-7),

f(x)=F_k^(-1)[F(k)](x)
(1)
=int_(-infty)^inftyF(k)e^(2piikx)dk,
(2)

where

F(k)=F_x[f(x)](k)
(3)
=int_(-infty)^inftyf(x)e^(-2piikx)dx
(4)

is the corresponding forward Fourier transform.

More generally, the inverse Fourier transform corresponding to FourierParameters -> {a, b} is

f(t)=sqrt((|b|)/((2pi)^(1+a)))int_(-infty)^inftyF(omega)e^(-ibomegat)domega.
(5)

The inverse Fourier transform of a function F(k) is implemented in the Wolfram Language as InverseFourierTransform[F, k, x]. By default, the Wolfram Language takes FourierParameters -> {0, 1}; in this work, following Bracewell (1999, pp. 6-7), it is always assumed that a=0 and b=-2pi unless otherwise stated.

See also

Fourier Transform, Integral Transform

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References

Bracewell, R. The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, 1999.Krantz, S. G. "The Fourier Transform." §15.2 in Handbook of Complex Variables. Boston, MA: Birkhäuser, pp. 202-212, 1999.

Referenced on Wolfram|Alpha

Inverse Fourier Transform

Cite this as:

Weisstein, Eric W. "Inverse Fourier Transform." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/InverseFourierTransform.html

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