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  "body": "\n## Definition\n\nLet $X$ be a set and let $f \\colon X \\to X$ be a given map.\nThe *Abel equation* for $f$ is the [[iterative functional equation|BCGBQ3HK26]]\n$$\\alpha(f(x)) = \\alpha(x) + 1 \\quad \\text{for all } x \\in X,$$\nfor an unknown [[function|0EPAT51DXQ]] $\\alpha \\colon X \\to \\R$ or $\\alpha \\colon X \\to \\C$.\nA solution is called an [[Abel function|2QBWT7T02Y]] of $f$.\n\n## Remarks\n\nExample: for $f(x) = x^2$ on $X = (1, \\infty)$, $\\alpha(x) = \\log_2 \\ln x$ is a solution, since $\\log_2 \\ln(x^2) = \\log_2(2 \\ln x) = \\log_2 \\ln x + 1$.\nNon-example: for $f(x) = 2x$ on $(0, \\infty)$, $\\alpha(x) = x$ is not a solution, since $\\alpha(f(2)) = 4$ while $\\alpha(2) + 1 = 3$.\nIf $\\alpha$ is a real-valued solution and $p \\colon \\R \\to \\R$ satisfies $p(t + 1) = p(t)$, then $\\alpha + p \\circ \\alpha$ is again a solution, so solutions are far from unique without extra conditions.\nIf $f$ has a fixed point $x_0$, no solution is defined at $x_0$, since $\\alpha(x_0) = \\alpha(x_0) + 1$ is impossible; so the equation is usually posed on a domain without fixed points.\n",
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