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  "body": "\n## Summary\n\nThe Koenigs function of a holomorphic map $f$ with an attracting, not superattracting, [[fixed point|R6QVS6P21N]] is the normalised solution of [[Schröder's equation|VTJV7JVJEA]] at that point.\nIt is the limit of the rescaled iterates $f^n(z) / \\lambda^n$, where $\\lambda = f'(0)$.\nIn its coordinate the map $f$ becomes multiplication by $\\lambda$; this is the content of the [[Koenigs linearization theorem|G02NBNN6CJ]].\nIt is a basic tool in complex dynamics and in the study of composition operators.\n\n## Explanations\n\n### L1 human\n\nIf a map pulls points towards $0$ and shrinks distances near $0$ by the factor $\\lambda$, then $n$ steps shrink by about $\\lambda^n$.\nDividing the $n$-th iterate by $\\lambda^n$ undoes this shrinking, and in the limit one gets a new coordinate.\nIn that coordinate the map is exactly multiplication by $\\lambda$.\nFor $f(z) = z / (2 - z)$ the factor is $\\lambda = 1/2$, and the Koenigs function is $z / (1 - z)$.\n",
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