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  "body": "\n## Definition\n\nLet $f(z) = \\sum_{n \\ge 0} a_n z^n$ be a transcendental [[entire function|0419794R0T]], that is, an entire function that is not a polynomial.\nWiman–Valiron theory studies $f$ through the following functions of $r \u003e 0$:\n\n1. the *maximum term* $\\mu(r) = \\max_{n \\ge 0} \\abs{a_n} r^n$;\n2. the *central index* $\\nu(r)$, the largest integer $n$ with $\\abs{a_n} r^n = \\mu(r)$.\n\n## Remarks\n\nThe maximum exists because $\\abs{a_n} r^n \\to 0$ as $n \\to \\infty$ for every $r$, by convergence of the [[Taylor series|0225QPWRRE]].\nThe function $\\nu(r)$ is nondecreasing, and $\\nu(r) \\to \\infty$ because $f$ is not a polynomial.\nExample: for $f = e^z$, $a_n = 1/n!$ and the ratio of consecutive terms is $r/(n+1)$, so $\\nu(r) = \\lfloor r \\rfloor$ and $\\mu(r) = r^{\\nu(r)}/\\nu(r)!$.\nNon-example: for a polynomial of degree $d$, $\\nu(r) = d$ for all large $r$, so it stays bounded; this is why the theory requires $f$ transcendental.\nThe Cauchy estimates give $\\mu(r) \\le M(r)$, where $M(r) = \\max_{\\abs{z} = r} \\abs{f(z)}$.\nThe main result states that, for $r$ outside a set $E$ with $\\int_E dr/r \u003c \\infty$ and $z_r$ with $\\abs{z_r} = r$ and $\\abs{f(z_r)} = M(r)$, one has $f^{(k)}(z_r) = \\left(\\nu(r)/z_r\\right)^k f(z_r)\\,(1 + o(1))$ as $r \\to \\infty$, for each fixed $k$.\n",
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