I would like to point out that tommysexp for bases \( >= e^2 \) is problemfree for real values.
The reason is that for bases \( >=e^2 \) there is a unique (entire) superfunction for the 2sinh ( example base \( e^2 \) thus \( e^{2x} - e^{-2x} \)).
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 \( >=e^2 \) there is a unique (entire) superfunction for the 2sinh ( example base \( e^2 \) thus \( e^{2x} - e^{-2x} \)).
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 ?