Fourier Transform of a Gaussian Density Function

In Spring 2026, a previous student of mine asked me to help deriving the steps inbetween the labeled equations on the following webpage: Fourier Transform--Gaussian.

We will connect equation (1)

to equation (4)

Surely, we could just take for granted Abramowitz and Stegun (1972, p. 302, equation 7.4.6), as is done on the webpage linked above. Instead, I'll supply my own attempt at filling in the blanks.

The tricks that I used are

• complete the square in the exponential,
• shifts of Gaussian densities don't change the integral, and
• Gaussian densities integrate to 1.

Starting from

add the exponents and factor out the .

Completing the square in gives

Remembering , we the completed square to be

Next, put this back into the integral

Factor out the constant (with respect to ) to get

Notice that inside the squared term is a constant with respect to . If you squint, this is like the location parameter of a Guassian density function. Recall, that for any value of , the value of the integral remains the same. So the above expression is equal to

Last, recall that the Gaussian density integrates to , so that

Notice the change of variables . Solve for , then make the appropriate substitution back.