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  "body": "\n## Summary\n\nThe order of an [[entire function|0419794R0T]] measures how fast its maximum modulus grows, on the scale of functions $\\exp(r^\\lambda)$.\nIt is the least exponent $\\rho$ such that $\\abs{f(z)}$ is at most about $\\exp(\\abs{z}^{\\rho})$ for large $\\abs{z}$.\nPolynomials have order $0$, $e^z$ has order $1$, and $e^{z^2}$ has order $2$.\nThe order controls the number of zeros of the function and the shape of its factorization, as in the [[Hadamard factorization theorem|KYSR7NHNFQ]].\n\n## Explanations\n\n### L1 human\n\nEntire functions can grow at very different speeds as $\\abs{z}$ grows.\nTo compare them, one looks at the largest value $M(r)$ of $\\abs{f}$ on the circle of radius $r$ and asks which power $r^\\rho$ appears in the exponent.\nFor $e^z$, $M(r) = e^r$, so the order is $1$.\nFor $\\cos \\sqrt{z}$ the growth is like $e^{\\sqrt{r}}$, so the order is $1/2$.\nA function such as $e^{e^z}$ grows faster than every $\\exp(r^\\rho)$ and has infinite order.\n",
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