Graduate Course: Quantum Physics

Mike Grams and John Venables, Dept of Physics and Astronomy, Arizona State University, Tempe, Arizona 

Fourier Transform of the Lorentzian

The Lorentzian function is


and the definition of the transform is

We can solve this integral by considering

where z = (x + iy) is a complex number.

This complex integral has poles at +a and -a, as shown in the diagram below. We can solve the integral by contour integration. The trajectory along the x-axis is what we want, the large circle goes to zero for large radius, R, and the residues around the pole is found by Cauchy's theorem.

If k > 0, the contour is closed in the lower half plane (as above), so the pole at z = - ia contributes. If k < 0, the contour is closed above the real axis, so the pole at z = + ia contributes.

For k < 0:due to a counterclockwise direction of contour:

This equals 2ip times

and the total integral becomes: . Similarly, for k > 0 (clockwise direction of contour) we obtain: . And finally the transform is:

The Lorentzian function and the transform look like this for alpha = 2. The narrower function is the transform.



Latest version of this document: 7th May 2000