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Fourier Transform--Gaussian

The Fourier transform of a Gaussian function f(x)=e^(-ax^2) is given by

F_x[e^(-ax^2)](k)=int_(-infty)^inftye^(-ax^2)e^(-2piikx)dx
(1)
=int_(-infty)^inftye^(-ax^2)[cos(2pikx)-isin(2pikx)]dx
(2)
=int_(-infty)^inftye^(-ax^2)cos(2pikx)dx-iint_(-infty)^inftye^(-ax^2)sin(2pikx)dx.
(3)

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

F_x[e^(-ax^2)](k)=sqrt(pi/a)e^(-pi^2k^2/a),
(4)

so a Gaussian transforms to another Gaussian.

See also

Gaussian Function, Fourier Transform

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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.

Cite this as:

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

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