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  "body": "\n## Statement\n\nLet $\\exp$ be the [[exponential function|0J0FCWV32D]].\nThere is a [[real-analytic|081WKEDK6C]], strictly increasing function $F \\colon (-2, \\infty) \\to \\R$ such that\n$$F(0) = 1 \\quad \\text{and} \\quad F(x + 1) = \\exp(F(x)) \\quad \\text{for all } x \u003e -2.$$\nSo $F$ extends [[tetration|NT65WHM5VV]] to base $e$ from integer heights to real heights: $F(n) = {}^{n}e$ for all integers $n \\ge 1$.\n\n## Remarks\n\nKneser's construction starts at a non-real fixed point $L$ of $\\exp$ in $\\C$, with $L \\approx 0.318 + 1.337i$.\nThere $\\exp'(L) = L$ has modulus greater than $1$, so $\\exp$ has a holomorphic Schröder function near $L$, and this gives a complex solution of the [[Abel equation|68SHB7Z721]].\nThis solution is not real on the real axis, and Kneser corrects it by a conformal change of variables, built with the Riemann mapping theorem, to obtain a real-analytic one.\nFrom $F$ one gets a real-analytic [[functional square root|E7890PHMGP]] of $\\exp$ on $\\R$: $g(x) = F\\big(F^{-1}(x) + \\tfrac12\\big)$ satisfies $g(g(x)) = \\exp(x)$.\nThe values at integers are forced: $F(-1) = \\log F(0) = 0$, $F(1) = e$, $F(2) = e^e$, and $F(x) \\to -\\infty$ as $x \\to -2$ from the right.\nThe equation $F(x + 1) = \\exp(F(x))$ with $F(0) = 1$ has many solutions, even smooth ones, so extra conditions are needed to single out Kneser's solution.\n\n## Explanations\n\n### L1 human\n\nTetration ${}^{n}e$ is a tower of $n$ copies of $e$, and it is easy to define when $n$ is a whole number.\nKneser showed that one can fill in the values between the whole numbers by a smooth curve, in fact an analytic one.\nThe curve still obeys the rule \"one more floor means one more exponential\".\nThis also gives a function that, applied twice, is the exponential function.\n\n### L2 human\n\nThe equation $F(x + 1) = \\exp(F(x))$ is an Abel-type equation for the inverse of $F$, and its solutions are easy to build piece by piece on intervals of length $1$.\nSuch solutions are usually not analytic, and the difficulty is to find a solution that is analytic everywhere.\nKneser's method combines the theory of linearization near a repelling fixed point, the Schröder and Koenigs theory, with the Riemann mapping theorem.\nIt is the standard reference point for later work on tetration to other bases and on uniqueness of analytic solutions.\n",
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