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  "body": "\n## Summary\n\nNevanlinna theory is the quantitative study of how often a [[meromorphic function|0NSD1TPHR0]] on $\\C$ takes each value.\nFor each value $a$ it uses a counting function, which counts the solutions of $f(z) = a$ in a disc.\nIt also uses a proximity function, which measures how close $f$ comes to $a$ on the boundary circle.\nIts two main theorems compare these quantities with a single growth measure, the [[Nevanlinna characteristic|YT6Z0GWS0W]].\nThe theory refines [[Picard's little theorem|R2PHM118NG]] and has analogues in Diophantine approximation.\n\n## Explanations\n\n### L1 human\n\nA nonconstant polynomial of degree $d$ takes every value exactly $d$ times.\nAn entire or meromorphic function can take a value infinitely often, or never, as $e^z$ never takes $0$.\nNevanlinna theory replaces the degree by a function $T(r, f)$ that grows with the radius $r$.\nThe [[first main theorem|QJWCFAK4AK]] says that each value is taken at most about $T(r, f)$ times, counted with a logarithmic weight.\nThe [[second main theorem|GWYE6745JK]] says that most values are taken about that often, and that at most two values can be strongly avoided.\n",
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