regular slog

[bo198214]

The Abel function has also a singularity at 0.

Just realized, this is only if the fixed point is 0.

otherwise at the fixed point. The regular iteration theory always assumes the fixed point at 0. If not one just considers the function \( f(x+a)-a \) where \( a \) is the fixed point.

\( FS=cS \)

This should be \( SF=cS \),

actually thats also wrong. However it is only an intermediate error in my derivation.
Lets show the correct equations:
\( \sigma(f(x))=c\sigma(x) \) or, with \( \mu_c(x)=cx \):
\( \sigma\circ f = \mu_c \circ \sigma \)
if we take the Bell matrices:
\( FS=SM \)
where \( M \) is the Bell matrix of \( \mu_c \). This is the diagonal matrix:
\( M=\begin{pmatrix}
c &0 & 0 &\dots &0\\
0 & c^2 & 0&\dots& 0\\
&&\vdots&\\
0 & &\dots& 0 & c^n\\
\end{pmatrix} \)
I think Gottfried calls this the Vandermonde matrix.

The right multiplication of this matrix multiplies each \( k \)-th column with \( c^k \). If we truncate \( S \) to its first column \( \vec{\sigma} \) we get hence:
\( F\vec{\sigma} = c\vec{\sigma} \)
and this can then be transformed to:
\( (F-cI)\vec{\sigma}=0 \)
which I used for my further derivations.