As I now have the fundamentals of computing the regular slog for a given fixed point, we can compute \( \alpha^\ast_{b,a}(x)=\frac{\alpha_{b,a}(x)+\alpha_{b,\overline{a}}(x)}{2}=\Re(\alpha_{b,a}(x)) \)
where \( \alpha_{b,a} \) is the principal Abel function of \( b^x \) at the fixed point \( a \) and I provide the graph of \( \alpha^\ast_{e,a} \) where \( a \) is the primary fixed point in the upper half plane of \( e^x \).
The result looks quite strange:
and is surely not Andrew's slog.
I mean there are to be expected kind of singularities at \( {^ne} \) which can well be seen. However another strange thing happens, the Abel function is not continuous at a strange point as we see when we extend the range:
I have no explanation yet.
where \( \alpha_{b,a} \) is the principal Abel function of \( b^x \) at the fixed point \( a \) and I provide the graph of \( \alpha^\ast_{e,a} \) where \( a \) is the primary fixed point in the upper half plane of \( e^x \).
The result looks quite strange:
and is surely not Andrew's slog.
I mean there are to be expected kind of singularities at \( {^ne} \) which can well be seen. However another strange thing happens, the Abel function is not continuous at a strange point as we see when we extend the range:
I have no explanation yet.
