# Never-before-seen images of the Lévy stable distribution

Analytic continuation to the complex plane of the Lévy stable distribution, as the Lévy index is varied from the Cauchy to the Gaussian limits.   We have used domain coloring to plot a complex function of complex numbers.

The $\alpha$-stable Lévy distribution has been known since the 1930s and been the subject of systematic and persistent study for many decades.  The most important special cases of the  Lévy stable distribution are the Gaussian distribution (also known as the normal distribution) and the Cauchy distribution (also known as the Lorentzian distribution).

My doctoral student Éric C. Rocha, as part of his thesis project, generated domain coloring plots of the analytic continuation to the complex plane of Lévy stable distribution.  Some of these images have been published in Physical Review E (see PDF file).

The animated GIF above shows the Lévy stable distribution as $\alpha$ varies from 1 to 2.   The $\alpha=1$ case is the Cauchy and the $\alpha=2$ case is the Gaussian. The complex conjugate poles on the imaginary axis are clearly visible for the Cauchy case. As the distribution is deformed towards the Gaussian, one can see a “palm leaf structure” form due to the presence of zeroes. Finally, the zeroes go off to infinity and the shape converges to the saddle structure of the Gaussian.

Enjoy!

I thank Wolfram for donating a licence for Mathematica.