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  "body": "\n## Statement\n\nLet $f$ be an [[entire function|0419794R0T]] of [[order|SHF8Z7GMFX]] $\\rho$.\nCall $a \\in \\C$ a finite asymptotic value of $f$ if there is a continuous curve $\\gamma \\colon [0, 1) \\to \\C$ with $\\gamma(t) \\to \\infty$ and $f(\\gamma(t)) \\to a$ as $t \\to 1$.\nIf $f$ has $n \\ge 1$ distinct finite asymptotic values, then $n \\le 2\\rho$.\n\n## Remarks\n\nBetween two curves along which $f$ tends to different finite limits, $f$ must be unbounded, so $f$ has at least $n$ separate regions near $\\infty$ where $\\abs{f}$ is large.\nAhlfors's proof bounds the growth of $f$ in such regions with an estimate for conformal maps of strip-like domains.\nThis estimate is now called the Ahlfors distortion theorem.\nIt is proved by the length–area method, which later grew into the theory of [[extremal length|HX72KT9D0G]].\nFor example, $e^z$ has order $1$ and the single finite asymptotic value $0$, reached along the negative real axis, so $1 \\le 2$.\nFor $n = 1$ the theorem says that an entire function of order less than $1/2$ has no finite asymptotic value.\n\n## Explanations\n\n### L1 human\n\nAn asymptotic value is a limit that the function approaches along some road to infinity.\nThe theorem says that an entire function cannot have many different such limits unless it grows fast.\nEach extra finite limit costs half a unit of order.\n\n### L2 human\n\nThe order is defined by $\\rho = \\limsup_{r \\to \\infty} \\log \\log M(r)/\\log r$, where $M(r)$ is the maximum of $\\abs{f}$ on $\\abs{z} = r$.\nThe case $n = 1$ is consistent with the $\\cos \\pi\\rho$ theorem, which shows that functions of order less than $1/2$ are large on a sequence of whole circles.\nThe method of the proof, through extremal length and conformal invariants, became a central tool of geometric function theory.\n",
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