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  "body": "\n## Definition\n\nLet $f$ be a [[meromorphic function|0NSD1TPHR0]] on $\\C$ that is not constant, and write $\\log^+ x = \\max(\\log x, 0)$ for $x \\ge 0$, with $\\log^+ 0 = 0$.\nNevanlinna theory studies $f$ through the following functions of $r \u003e 0$, for $a$ in the [[Riemann sphere|0CDMXGPSES]] $\\C \\cup \\set{\\infty}$.\n\n1. 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.\n2. 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 $\\overline{n}$.\n3. 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 $m(r, \\infty) = \\dfrac{1}{2\\pi} \\displaystyle\\int_0^{2\\pi} \\log^+ \\abs{f(re^{i\\theta})} \\, d\\theta$.\n4. The [[Nevanlinna characteristic|YT6Z0GWS0W]] $T(r, f) = m(r, \\infty) + N(r, \\infty)$.\n5. The *defect* $\\delta(a) = \\liminf_{r \\to \\infty} \\dfrac{m(r, a)}{T(r, f)}$, using the [[limit inferior|0H4R3RESGV]].\n\n## Remarks\n\nExample: for $f(z) = e^z$ there are no poles and no zeros, so $N(r, \\infty) = N(r, 0) = 0$.\nThe proximity functions are $m(r, \\infty) = m(r, 0) = \\frac{1}{2\\pi}\\int_0^{2\\pi} \\max(r \\cos\\theta, 0) \\, d\\theta = r/\\pi$, so $T(r, e^z) = r/\\pi$ and $\\delta(0) = \\delta(\\infty) = 1$.\nFor a value $a \\ne 0, \\infty$, the solutions of $e^z = a$ have $n(t, a) \\sim t/\\pi$, so $N(r, a) \\sim r/\\pi$ and $\\delta(a) = 0$.\nBy the [[first main theorem|QJWCFAK4AK]], $\\delta(a) = 1 - \\limsup_{r \\to \\infty} N(r, a)/T(r, f)$ when $f$ is transcendental, and $0 \\le \\delta(a) \\le 1$.\nA value that $f$ omits has defect $1$.\n",
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