Calculus and Analysis > Integral Transforms > Fourier Transforms >

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.

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