bo198214 Wrote:I can make a more concrete reply

The "more concrete reply" (and the maple code that calculate the coefficients), is at

http://math.eretrandre.org/tetrationforu...hp?tid=215
Here I continue about branchpoints. Cutlines of the superlogarithm transfer to cuts of

. Therefore, if one is interested to look behind the cutlines,

first, first look, that is happening with slog. I attach the contourplot of real and imaginary parts of slog in the left figure. It was already posted there, but now I post it with impeover desolution. At the right hand side, the similar figure with cut moved down.

[

attachment=429]

Technically, for moving the cuts down, instead of slog, I plot the function

defined with

.

In this case, the cutlines remain straight, but the additional cutline along the negative part of the real axis appears.

Define the new operation cutter, which maked from a function

function

In the right hand side of the picture above, cutter(slog) is plotted. Applying this cutter sequentially, one may cut the complex plane to the upper part and the lower part, which are holomorphically connected only at the right hand side of the real axis.

In the similar way, define the operation shroeder with equation

. In the two pictures below I plot

and

in the same notations.

[

attachment=430]

The schroeder cuts the (simple) complex plane to the complicated (really "complex") labirinth with walls created by the cutlines. The cutlines (and singularities) of the sexp also contribute to the cutlines (and singulatities) of the generalized exponential.