I would like to point out that tommysexp for bases is problemfree for real values.

The reason is that for bases there is a unique (entire) superfunction for the 2sinh ( example base thus ).

And that is true because 2sinh(2x) has only one complex fixpoint : 0.

So for these bases tommysexp is valid , has uniqueness and existance and satisfies all desired properties with the possible exception of analytic. However it is Coo.

While doing basechange I prefer to use tommysexp base exp(2).

Lets call that basechangetommy.

I suspect basechangetommy and tommysexp for bases larger than exp(2) to have similar properties.

I even considered them to be equal once , although I now keep it as a sort of vague closeness idea.

I am considering properties and might even call it my new pet ideas in tetration.

I believe in intresting properties for these functions.

Im also considering replacing the logs with functions that are asymptotic to logs but entire. ( an idea that I came up with together with mick , the guy who posts on MSE )

However that is quite complicated and not well understood at the moment. For instance it is unknown if this is the sought of analytic continuation or a totally different function ?

The reason is that for bases there is a unique (entire) superfunction for the 2sinh ( example base thus ).

And that is true because 2sinh(2x) has only one complex fixpoint : 0.

So for these bases tommysexp is valid , has uniqueness and existance and satisfies all desired properties with the possible exception of analytic. However it is Coo.

While doing basechange I prefer to use tommysexp base exp(2).

Lets call that basechangetommy.

I suspect basechangetommy and tommysexp for bases larger than exp(2) to have similar properties.

I even considered them to be equal once , although I now keep it as a sort of vague closeness idea.

I am considering properties and might even call it my new pet ideas in tetration.

I believe in intresting properties for these functions.

Im also considering replacing the logs with functions that are asymptotic to logs but entire. ( an idea that I came up with together with mick , the guy who posts on MSE )

However that is quite complicated and not well understood at the moment. For instance it is unknown if this is the sought of analytic continuation or a totally different function ?