Simultaneous deficiency and ramification for an exponential polynomial
= Simultaneous deficiency and ramification for an exponential polynomial
{title2=$f=e^z(e^z-1)^2,\quad\delta_f(0)=\theta_f(0)=1/3$}
The <rational composition law for the Nevanlinna characteristic> gives $T_f(r)=3r/\pi+O(1)$. Its zeros are precisely $2\pi i\mathbb Z$, all of <multiplicity> two, so $N(r;0)=2r/\pi+O(\log r)$ and $\overline N(r;0)=r/\pi+O(\log r)$. Hence both its <Nevanlinna deficiency> and <Nevanlinna ramification index> at zero equal one third.