Lorentzian Function

The Lorentzian function is given by

$$L(x)={1\over\pi} {{\textstyle{1\over 2}}\Gamma\over(x-x_0)^2+({\textstyle{1\over 2}}\Gamma)^2}.$$

Its Full Width at Half Maximum is $\Gamma$. This function gives the shape of certain types of spectral lines and is the distribution function in the Cauchy Distribution. The Lorentzian function has Fourier Transform

$${\mathcal F}\left[{{1\over\pi} {{\textstyle{1\over 2}}\Gamma... ...r 2}}\Gamma)^2}}\right]=e^{-2\pi ikx_0-\Gamma\pi\vert k\vert}.$$

See also Damped Exponential Cosine Integral, Fourier Transform--Lorentzian Function