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  "body": "\n## Definition\n\nLet $(a_n)_{n \\ge 0}$ be a sequence of counts with [[generating function|0PHAC833JF]] $A(z) = \\sum_{n} a_n z^n$ or $\\sum_n a_n z^n / n!$.\n*Analytic combinatorics* is the branch of [[enumerative combinatorics|EX382NQTGP]] that derives the asymptotic behaviour of $a_n$ as $n \\to \\infty$ from $A$.\nHere $A$ is viewed as an [[analytic function|081WKEDK6C]] of a complex variable $z$.\nIts main methods are:\n\n1. location of the dominant singularities, which fixes the exponential growth rate;\n2. singularity analysis, which transfers the local behaviour of $A$ near a dominant singularity to the coefficients;\n3. saddle-point estimates of the Cauchy coefficient integral, mainly for entire functions $A$.\n\n## Remarks\n\nThe basic principle is the Cauchy–Hadamard formula.\nIf $A$ has [[radius of convergence|0N288P2TB5]] $\\rho$ with $0 \u003c \\rho \u003c \\infty$, then $\\limsup_{n \\to \\infty} \\abs{a_n}^{1/n} = 1/\\rho$.\nWhen $a_n \\ge 0$, Pringsheim's theorem says that $z = \\rho$ is itself a singularity of $A$.\n\nExample: the Catalan generating function is $C(z) = \\frac{1 - \\sqrt{1 - 4z}}{2z}$.\nIts dominant singularity is $z = 1/4$, where $C$ behaves like $2 - 2\\sqrt{1 - 4z}$.\nSingularity analysis then gives $C_n \\sim \\frac{4^n}{\\sqrt{\\pi}\\, n^{3/2}}$.\n\nNon-example: a proof that $\\binom{2n}{n} \\le 4^n$ by counting subsets is not analytic combinatorics.\nIt uses no property of a generating function as a function of a complex variable.\n",
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