Generalised Cauchy Distribution

The function I discovered a few years ago turns out, on re-examination, to be a generalised Cauchy Distribution:

Where x is the variable value, and parameters ‘m’ is mode, ‘s’ controls fatness, and ‘a’ controls kurtosis. I assume, like Cauchy, it has no moments, but need to explore.

With a = 1, s = 1 it gives the Cauchy Distribution :

 

With a = 2 it gives a useful distribution with fat tails, fatter body and with mode similar to Normal and LogNormal. ‘sigma’ appears to be derivable from standard deviation here!

This has a Cumulative Distribution :

 

If C( x) is the Cauchy Distribution then GC2(x;0,1) = 2 Pi C^2