04/23/2014, 09:15 PM

Forgive me if this is an old question or it could be easily derived from previous posts. Or if my memory fails me.

But despite many posts on slog_b(z) and sexp_b(z) , I am somewhat puzzled by slog_b(sexp_b(z)).

For clarity b is the base and in particular I am (mainly) intrested in bases larger than exp(1/2).

Also to avoid confusion slog_b(sexp_b(z)) =/= z.

This is the tetration analogue of log_b(exp_b(z)) =/= z.

As often in the topic of tetration I feel " lured into deception ".

By that I mean that many ideas pop up naturally but none of them seem very solid , intresting or true in retrospect.

Also many variations on this question occur naturally , usually in the form of special cases and restrictions.

Such as Re(z) > 0.

Looking at changing values of b , rather than say z.

Or considering different solutions to sexp and how that afffects things.

A few notes :

1) analytic continuation is problematic since for REAL z , the equation does in fact hold ! ( and continuation of z = z of course ).

This also makes me distrust Taylor series and related calculus.

2) my 2sinh fails here or is meaningless since it is not analytic !

Well at least I think so , because it is then imho not well defined for nonreal imput (z).

3) the chaotic nature of interations z , exp(z) , exp^[2](z) , ... is one of the reason I doubt my own intuition and I feel " lured into deception ".

4) I could probably link or rewrite almost all threads , questions and answers as a note here , since I find that most threads relate to this in one way or another !

5) One of the " deceptions " was a " nonentire analytic function with all singularies at oo " what makes no sense of course. I however have no systematic way of doing things that automatically avoids such " nonsense ".

Maybe Im not the only one who wonders about this.

Some of my friends believe this requires new special functions to fully understand.

I considered using higher dimensional numbers but that seems like running before you can walk. So I guess the complex plane is the place to start.

regards

tommy1729

But despite many posts on slog_b(z) and sexp_b(z) , I am somewhat puzzled by slog_b(sexp_b(z)).

For clarity b is the base and in particular I am (mainly) intrested in bases larger than exp(1/2).

Also to avoid confusion slog_b(sexp_b(z)) =/= z.

This is the tetration analogue of log_b(exp_b(z)) =/= z.

As often in the topic of tetration I feel " lured into deception ".

By that I mean that many ideas pop up naturally but none of them seem very solid , intresting or true in retrospect.

Also many variations on this question occur naturally , usually in the form of special cases and restrictions.

Such as Re(z) > 0.

Looking at changing values of b , rather than say z.

Or considering different solutions to sexp and how that afffects things.

A few notes :

1) analytic continuation is problematic since for REAL z , the equation does in fact hold ! ( and continuation of z = z of course ).

This also makes me distrust Taylor series and related calculus.

2) my 2sinh fails here or is meaningless since it is not analytic !

Well at least I think so , because it is then imho not well defined for nonreal imput (z).

3) the chaotic nature of interations z , exp(z) , exp^[2](z) , ... is one of the reason I doubt my own intuition and I feel " lured into deception ".

4) I could probably link or rewrite almost all threads , questions and answers as a note here , since I find that most threads relate to this in one way or another !

5) One of the " deceptions " was a " nonentire analytic function with all singularies at oo " what makes no sense of course. I however have no systematic way of doing things that automatically avoids such " nonsense ".

Maybe Im not the only one who wonders about this.

Some of my friends believe this requires new special functions to fully understand.

I considered using higher dimensional numbers but that seems like running before you can walk. So I guess the complex plane is the place to start.

regards

tommy1729