the Nevanlinna characteristic function
/ Nevanlinna -> NEH-vahn-lin-nah /
Growth theory for entire functions has a natural ruler in log M(r), the log of the maximum modulus. But meromorphic functions have poles where |f| blows up, so M(r) is useless for them, and even for entire functions the maximum is a blunt, one-direction measurement. The Nevanlinna characteristic function T(r) is the better ruler — a single increasing function of r that gauges the total growth of a meromorphic function in a balanced, averaged way, and serves as the universal yardstick of the whole value-distribution theory.
T(r) is built from two halves: T(r) = m(r) + N(r). The first half, the proximity term m(r), is the average of the positive part of log|f| over the circle |z| = r — precisely m(r) = (1 / 2 pi) times the integral of max(log|f(r e^(i theta))|, 0) d theta. It records how big f gets on the boundary, on average, where it is large. The second half, the counting term N(r), is the integrated count of the poles of f inside radius r (the same N as the zero-counting function, but for poles). Together they capture both ways f can be 'large': by taking big values (m) and by having poles (N). For an entire function there are no poles, N for poles is zero, and T(r) reduces to the proximity term, which turns out to be comparable to log M(r) — so T(r) genuinely generalizes the old ruler.
The characteristic function is the centerpiece because every quantity in Nevanlinna theory is measured against it: the order is rho = limsup of log T(r) / log r, the First Main Theorem says m(r, w) + N(r, w) = T(r) + O(1) for every value w (so T(r) is the shared budget), and the deficiencies are defined as ratios against T(r). Its great virtues are that it is essentially independent of the target value w and that it behaves well under arithmetic — T(r) for f + g and f times g are controlled by the sum of the individual T's. The caveat to remember: T(r) is defined only up to a bounded additive error and an averaging convention, so it is the GROWTH ORDER of T(r), not its exact value, that carries the meaning; two reasonable definitions of T(r) differ by O(1) and give the same theory.
For a rational function f = P/Q with deg P = p and deg Q = q, the characteristic grows logarithmically: T(r) is about max(p, q) times log r. This matches the intuition that rational functions are the 'order 0' meromorphic functions. For a transcendental entire function like e^z, by contrast, T(r) grows like a power of r (here like r), and the ratio log T(r) / log r tends to the order 1.
T(r) grows like log r for rational functions and like a power of r for transcendental ones — it reads off the order.
T(r) is defined only up to a bounded O(1) term and depends on an averaging convention, so its meaning lies in its growth rate, not its exact value. For entire functions T(r) is comparable to log M(r), but for general meromorphic functions the pole-counting term N(r) is essential and M(r) would be infinite.