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...an [[Abel function]] of a function that is 'disturbed' by some small $c\in\C$: $$f_c(z)=c + z + c_{m+1} z^{m+1} + o(z^{m+1})$$

For a given function $f$ and constant $c$ find a function $\sigma$ such that $$\sigma(f(x))=c \sigma(x).$$

$$\phi_c(x) = \alpha^{-1}(c+\alpha(x)), c\in\Q$$

Our concern here are functions of the form $f(z)=\frac{az+b}{cz+d}$ with $c\neq 0$. In the case $c\neq 0$, the fixpoints are given by

...b>e^{1/e}$ there exists exactly one holomorphic function $\sigma_b$ on $D=\C\setminus\{x \le -2\}$ which is real for all $x>-2$ and satisfies for all $z ...r all $k\ge 1$. Hence $\delta$ is a constant $c$ and $\sigma_b(z)=\tau_b(z+c)$.

:<math>(C[b^x]^T - I)D[\alpha] = D[1]\,</math> ...lor coefficients of the Abel function <math>\alpha(x)</math> making <math>(C[b^x]^T - I)</math> invertible (by truncating the first column and last row)

...of the other. A neighborhood of an analytic map $f\colon D(f)\to\overline{\C}$ is a set containing For $b_1,b_2\in\C$ with $\Re(b_1)<\Re(b_2)$ define

$x\in\R$ huhu and $z\in\C$