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  "body": "\n## Definition\n\nLet $U \\subseteq \\C$ be an open set containing $0$, and let $f \\colon U \\to \\C$ be [[holomorphic|0NAHXR1FRH]] with $f(0) = 0$.\nLet $\\lambda = f'(0)$.\n*Poincaré's functional equation* for $f$ is the [[functional equation|SRAN4EKWS0]]\n$$\\Psi(\\lambda z) = f(\\Psi(z)) \\quad \\text{for all } z \\text{ near } 0.$$\nThe unknown is a holomorphic function $\\Psi$ on a neighbourhood of $0$ with $\\Psi(0) = 0$ and $\\Psi'(0) = 1$.\n\n## Remarks\n\nExample: $f(w) = w^2 + 2w$ has $f(0) = 0$ and $\\lambda = f'(0) = 2$.\nThe function $\\Psi(z) = e^z - 1$ satisfies $\\Psi(0) = 0$, $\\Psi'(0) = 1$ and $f(\\Psi(z)) = (e^z - 1)^2 + 2(e^z - 1) = e^{2z} - 1 = \\Psi(2z)$.\n\nNon-example: for the same $f$, the function $\\Psi(z) = z$ fails, since $f(z) = z^2 + 2z \\ne 2z$ for $z \\ne 0$.\n\nIf $\\Phi$ is a solution of [[Schröder's equation|VTJV7JVJEA]] $\\Phi \\circ f = \\lambda \\Phi$ with $\\Phi(0) = 0$ and $\\Phi'(0) = 1$, then its local inverse $\\Psi = \\Phi^{-1}$ solves Poincaré's equation.\nSo for $\\abs{\\lambda} \u003e 1$ a unique solution exists, by the Koenigs linearization theorem applied to $f$.\nAssume moreover that $f$ is [[entire|0G4F0XB9X2]].\nThen the formula $\\Psi(z) = f^n(\\Psi(z / \\lambda^n))$, with $n$ so large that $z / \\lambda^n$ lies in the domain of $\\Psi$, extends $\\Psi$ to an entire function.\nThis function is the Poincaré function of $f$ at $0$.\n",
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