It also extends the usual range of scale parameter to include σ < 0.
An alternative form for the density can be written using the complex parameter θ = μ + iσ as where
To the question "Why introduce complex numbers when only real-valued random variables are involved?
", McCullagh wrote: To this question I can give no better answer than to present the curious result that
with respect to real linear fractional transformations (group action of SL(2,R)), and show that all f-divergences between univariate Cauchy densities are symmetric.