\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(q\right) & = & \tan\left(\pi q-\frac{\pi}{2}\right)\\
m_{d} & = & 0\\
m_{n} & = & 0\end{eqnarray*}

\begin{eqnarray*} h\left[X\right] & = & \log\left(4\pi\right)\\  & \approx & 2.5310242469692907930.\end{eqnarray*}