If we apply the FMT of nevanlinna's theory to $f(z)=e^z$ we get $$m_f(r,1)+N_f(r,1)=T_f(r)+O(1)$$ It is then easy to see that $m_f(r,1)=o(1); N_f(2\pi n,1)=(2n+1)\ln(n)-2\ln(n!); T_f(r)=\frac r\pi$. From this we get
This is almost Stirling's formula; moreover, once one has proved that $O(1)=cost+o(1)$, the constant can easily be deduced from Wallis's product. Is there a way to obtain from this line of argument the full Stirling formula?