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  "body": "\n## Definition\n\nLet $f$ be a [[meromorphic function|0NSD1TPHR0]] on $\\C$, not identically zero, and write $\\log^+ x = \\max(\\log x, 0)$ for $x \\ge 0$, with $\\log^+ 0 = 0$.\nFor $t \\ge 0$, let $n(t)$ be the number of [[poles|0WPQYW8MTE]] of $f$ in the disc $\\abs{z} \\le t$, counted with multiplicity.\nFor $r \u003e 0$, let\n$$m(r, f) = \\frac{1}{2\\pi} \\int_0^{2\\pi} \\log^+ \\abs{f(r e^{i\\theta})} \\, d\\theta, \\qquad N(r, f) = \\int_0^r \\frac{n(t) - n(0)}{t} \\, dt + n(0) \\log r.$$\nThe *Nevanlinna characteristic* of $f$ is the function\n$$T(r, f) = m(r, f) + N(r, f), \\qquad r \u003e 0.$$\n\n## Remarks\n\nExample: for $f(z) = e^z$, $\\log^+ \\abs{f(re^{i\\theta})} = \\max(r\\cos\\theta, 0)$ and there are no poles, so $T(r, f) = \\frac{1}{2\\pi}\\int_{-\\pi/2}^{\\pi/2} r \\cos\\theta \\, d\\theta = r/\\pi$.\nNon-example: the maximum modulus $\\max_{\\abs{z} = r} \\abs{f(z)}$ is no growth measure for $f(z) = \\tan z$.\nIt is infinite whenever the circle $\\abs{z} = r$ passes through a pole $\\pi/2 + k\\pi$.\nThe characteristic avoids this by counting the poles in $N(r, f)$.\nThe function $T(r, f)$ is increasing and a convex function of $\\log r$.\nFor an entire function $f$ and $M(r) = \\max_{\\abs{z} = r} \\abs{f(z)}$, one has $T(r, f) \\le \\log^+ M(r) \\le 3\\, T(2r, f)$, so $T$ and $\\log M$ define the same [[order|SHF8Z7GMFX]].\nA meromorphic function is rational exactly when $T(r, f) = O(\\log r)$, and a rational function of degree $d$ has $T(r, f) = d \\log r + O(1)$.\nThe Ahlfors–Shimizu characteristic, defined from the spherical area of the image of the disc $\\abs{z} \u003c r$, differs from $T(r, f)$ by a bounded function.\n",
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