Fourier Transform--Gaussian
The Fourier transform of a Gaussian function f(x)=e^(-ax^2) is given by
F_x[e^(-ax^2)](k) = [画像:int_(-infty)^inftye^(-ax^2)e^(-2piikx)dx]
(1)
![画像:int_(-infty)^inftye^(-ax^2)[cos(2pikx)-isin(2pikx)]dx](/index.cgi/larger-text/https://mathworld.wolfram.com/images/equations/FourierTransformGaussian/Inline7.svg){kind=link}
The second integrand is odd, so integration over a symmetrical range gives 0. The value of the first integral is given by Abramowitz and Stegun (1972, p. 302, equation 7.4.6), so
=sqrt(pi/a)e^(-pi^2k^2/a),](/index.cgi/larger-text/https://mathworld.wolfram.com/images/equations/FourierTransformGaussian/NumberedEquation1.svg){kind=link}
See also
Gaussian Function, Fourier TransformExplore with Wolfram|Alpha
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References
Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 302, 1972.Bracewell, R. The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, pp. 98-101, 1999.Referenced on Wolfram|Alpha
Fourier Transform--GaussianCite this as:
Weisstein, Eric W. "Fourier Transform--Gaussian." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/FourierTransformGaussian.html