[MSE] short review/implem. of Andy's method and a next step

[Gottfried]

Hi,   
just a couple of days ago I felt obliged to answer one question in MSE, where some member showed his conditions & equations coming from analysis of the tetration-mathematics.            
After some detective/investigation I understood that he attempted to set up an Abel-equation.      

To give him an impression of a way to solve it for the coefficients of the Abel-function (and its inverse) I "reengineered" Andy Robbins' method and showed some values from his slog() function. 

However, I went one step further and calculated the inverse powerseries for his slog() (where the constant term was removed), and surprisingly found a nice function for the tetration: \[\phi(x) = \,^x e\]
with a nice computational behave. 

It is a long text, but if this would be interesting for our forum, I could transfer this answer to our forum, may be at one of Andydude's JayDFox's threads where they explained their method (hints for a good place are then welcome).

Gottfried

[andydude]

Gottfried, my friend! How many years has it been?

My life is very different from when I joined this forum. I moved several times. Switched jobs, friends, outlook, and never seemed to have time for my passion for mathematics amongst the hustle and bustle of life. I just recently moved, and while sorting through old papers, I found a printout of my compendium of iteration stuff, noting that most of the chapters were still in bare bones outline form, which got me thinking about my original goal. To have better documentation, better libraries, and to do a better job of welcoming people into the community. I think we were pretty close on the computation front, getting sage and Pari code working, but it never had that "welcome to the future" look and feel that I would expect.

I have no idea about which thread to post in, since I've been a hermit for far too long, but I would love to help you convert whatever you have into something that the community can use.

[Catullus]

Welcome back andydude!
It has been a while.

[Daniel]

Gottfried, my friend! How many years has it been?

My life is very different from when I joined this forum. I moved several times. Switched jobs, friends, outlook, and never seemed to have time for my passion for mathematics amongst the hustle and bustle of life. I just recently moved, and while sorting through old papers, I found a printout of my compendium of iteration stuff, noting that most of the chapters were still in bare bones outline form, which got me thinking about my original goal. To have better documentation, better libraries, and to do a better job of welcoming people into the community. I think we were pretty close on the computation front, getting sage and Pari code working, but it never had that "welcome to the future" look and feel that I would expect.

I have no idea about which thread to post in, since I've been a hermit for far too long, but I would love to help you convert whatever you have into something that the community can use.

Greeting Andydude, thanks for checking in. From time to time I wonder how you and other have been doing. Good to have you back.

[Gottfried]

Hi Andrew - happy to see you back! Just this short note, I'll consider your infos & ideas when I'm finished with home duties :-) 

Welcome back!

Gottfried