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  "body": "\n## Summary\n\nThe Nevanlinna characteristic $T(r, f)$ of a [[meromorphic function|0NSD1TPHR0]] $f$ on $\\C$ measures its growth on the disc of radius $r$.\nIt adds a term for the size of $f$ on the circle $\\abs{z} = r$ and a term that counts the poles inside.\nFor entire functions it plays the role of $\\log M(r)$, and for meromorphic functions it replaces the degree of a rational function.\nIt is the central quantity of [[Nevanlinna theory|1EHH63VNAD]].\n\n## Explanations\n\n### L1 human\n\nFor an entire function, growth is measured by the maximum of $\\abs{f}$ on circles.\nA meromorphic function has poles, where it is infinite, so the maximum is useless.\nNevanlinna's idea is to average $\\log \\abs{f}$ where it is large on the circle and to add a logarithmic count of the poles inside.\nFor a rational function of degree $d$, $T(r, f)$ is about $d \\log r$.\nFor $e^z$, $T(r, f) = r/\\pi$.\n",
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