bo198214 Wrote:...Let \( D=\mathbb{C}\setminus [x_0,\infty) \) for some \( x_0<x_1 \)...Henryk: For tetration, I would define
\( D=\mathbb{C}\setminus (-\infty, x_0] \) for some \( x_0<x_1 \)...
then I like your proof.
I include below the small part of
http://math.eretrandre.org/tetrationforu...77#pid2477
which is picture of slog(S), assuming, \( x_1=-1 \) and that \( \epsilon \) is small and not seen. Vertical lines correspond to
\( \Re(\mathrm{sexp}(z))=0 \) and
\( \Re(\mathrm{sexp}(z))=1 \)
Horisontal lines correspond to
\( \Im(\mathrm{sexp}(z))=0 \) and
\( \Im(\mathrm{sexp}(z))=1 \)
The curvilinear mesh is produced by images of lines
\( \Re(z)= 0.2, 0.4, 0.6, 0.8, 1 ; \Im(z)>0 \) and
\( \Im(z)= 0.2, 0.4, 0.6, 0.8, 1, 1.2, 1.4, 1.6, 1.8, 2, 2.2, 2.4, 2.6, 2.8, 4; 0\le /Re(z) \le 1 \).
The pink cutline corresponds to \( \Re(z)= -2 \).
The images of lines with integer \( \Im(z) \) are a little bit extended.
Also, images of lines \( \Re(z)=2 \) and \( \Re(z)=3 \)are shown.
There is biholomorphizm \( \mathrm{sexp}(S): \leftrightarrow S \).
Henrik,
1. Do you plan to polish this proof or I may include this into the paper?
2. Can we claim, that some of singularities of a modified tetration are at
\( \Re(z)>-2 \) ? (In this case, we can include the case \( 1<b<\exp(1/{\mathrm e}) \) at once).