Exponential-Lorentzian FT

The Fourier transform of an exponential function, G(x) is a Lorentzian function, H(k). We illustrate this with the position wavevector conjugate pair.

Substituting in the value of G(x) = exp{-|x|/a} we have

The integral is exponential:

We use a substitution.

The integral is -1 so we have

Multiply both top and bottom by the complex conjugate.

The result shows that the Fourier transform of an exponential is a Lorentzian. For now we consider only the real part. As seen in the Figure below, the narrower the distribution in x space (exponential function), the broader it is in k space (Lorentzian function).

The conjugate relationship between the x-space and k-space is evident in the two Figures above.