Fourier Transform of a Scaled and Shifted Gaussian

On this page, we'll make use of the shifting property and the scaling property of the Fourier Transform to obtain the Fourier Transform of the scaled Gaussian function given by: [Equation 1] In Equation [1], we must assume K>0 or the function g(z) won't be a Gaussian function (rather, it will grow without bound and therefore the Fourier Transform will not exist). To start the process of finding the Fourier Transform of [1], let's recall the fundamental Fourier Transform pair, the Gaussian: [Equation 2] Let's first define the function h(z): [Equation 3] Observe that we have defined the constant c=sqrt( 4*pi*K ). We can relate the function h(z) and n(z) by the simple relation: h(z)=n(cz). Since we know the Fourier Transform of n(z) (Equation [2]), we can use the scaling property of the Fourier Transform to get the Fourier Transform of h(z): [Equation 4] In Equation [4], we have assumed K (and hence c) is positive. To find G(f), the Fourier Transform of g(z), we note that g(z) = h(z-a), and use the shift property of the Fourier Transform: [Equation 5] And thus, we have found the Fourier Transform of Equation [1]! Have a good day everybody.