Cauchy DistributionΒΆ
\[ \begin{eqnarray*} f\left(x\right) & = & \frac{1}{\pi\left(1+x^{2}\right)}\\ F\left(x\right) & = & \frac{1}{2}+\frac{1}{\pi}\tan^{-1}x\\ G\left(\alpha\right) & = & \tan\left(\pi\alpha-\frac{\pi}{2}\right)\\ m_{d} & = & 0\\ m_{n} & = & 0\end{eqnarray*}\]
No finite moments. This is the t distribution with one degree of freedom.
\[ \begin{eqnarray*} h\left[X\right] & = & \log\left(4\pi\right)\\ & \approx & 2.5310242469692907930.\end{eqnarray*}\]
Implementation: scipy.stats.cauchy