Nevanlinna theory

Definition

Let $f$ be a meromorphic function on $\C$ that is not constant, and write $\log^+ x = \max(\log x, 0)$ for $x \ge 0$, with $\log^+ 0 = 0$. Nevanlinna theory studies $f$ through the following functions of $r > 0$, for $a$ in the Riemann sphere $\C \cup \set{\infty}$.

1. The number $n(t, a)$ of solutions of $f(z) = a$ in the disc $\abs{z} \le t$, counted with multiplicity; for $a = \infty$ these are the poles of $f$. The number $\overline{n}(t, a)$ counts the same points without multiplicity.
2. The counting function $N(r, a) = \int_0^r \dfrac{n(t, a) - n(0, a)}{t} \, dt + n(0, a) \log r$, and $\overline{N}(r, a)$ defined in the same way from $\overline{n}$.
3. The proximity function $m(r, a) = \dfrac{1}{2\pi} \displaystyle\int_0^{2\pi} \log^+ \frac{1}{\abs{f(re^{i\theta}) - a}} \, d\theta$ for $a \in \C$, and $m(r, \infty) = \dfrac{1}{2\pi} \displaystyle\int_0^{2\pi} \log^+ \abs{f(re^{i\theta})} \, d\theta$.
4. The Nevanlinna characteristic $T(r, f) = m(r, \infty) + N(r, \infty)$.
5. The defect $\delta(a) = \liminf_{r \to \infty} \dfrac{m(r, a)}{T(r, f)}$, using the limit inferior.