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  "body": "\n## Definition\n\nLet $K \\subseteq \\C$ be a [[compact set|00R0X1WGPX]] with more than one point such that $\\widehat{\\C} \\setminus K$ is simply connected, where $\\widehat{\\C} = \\C \\cup \\set{\\infty}$ is the [[Riemann sphere|0CDMXGPSES]].\nLet $\\Phi$ be the [[conformal map|0VHNKP1N3K]] of $\\widehat{\\C} \\setminus K$ onto $\\set{w : \\abs{w} \u003e 1} \\cup \\set{\\infty}$ with $\\Phi(\\infty) = \\infty$ and\n$$\\Phi(z) = \\frac{z}{c} + d_0 + \\frac{d_1}{z} + \\frac{d_2}{z^2} + \\cdots \\quad \\text{near } \\infty,$$\nwith $c \u003e 0$.\nFor $n \\ge 0$, the *$n$-th Faber polynomial* $F_n$ of $K$ is the polynomial part of the [[Laurent series|0X3XD2ATB0]] of $\\Phi(z)^n$ at $\\infty$.\nThat is, $F_n$ is the unique polynomial such that $\\Phi(z)^n - F_n(z) \\to 0$ as $z \\to \\infty$.\n\n## Remarks\n\nThe map $\\Phi$ exists and is unique by the [[Riemann mapping theorem|02WHMEG5ZR]], since $\\widehat{\\C} \\setminus K$ is simply connected and has more than one boundary point.\nThe constant $c$ is the [[logarithmic capacity|DBQXRTC0VN]] of $K$.\nThe polynomial $F_n$ has degree $n$ and leading coefficient $c^{-n}$.\nExample: for the disc $\\abs{z - a} \\le r$, $\\Phi(z) = (z - a)/r$ and $F_n(z) = ((z - a)/r)^n$.\nExample: for $K = [-1, 1]$, $\\Phi(z) = z + \\sqrt{z^2 - 1}$, and $\\Phi^n + \\Phi^{-n} = 2 T_n$ with $T_n$ the Chebyshev polynomial; since $\\Phi^{-n} \\to 0$ at $\\infty$, $F_n = 2 T_n$ for $n \\ge 1$.\nNon-example: the union of two disjoint closed discs does not qualify, because its complement in $\\widehat{\\C}$ is not simply connected, so there is no map $\\Phi$.\nWith $\\Psi = \\Phi^{-1}$, the Faber polynomials have the generating function $\\Psi'(w)/(\\Psi(w) - z) = \\sum_{n \\ge 0} F_n(z) w^{-n-1}$ for $\\abs{w}$ large.\n",
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