(06/02/2011, 10:27 PM)sheldonison Wrote: There is also a lower superfunction for base eta, that I usually refer to as \( \text{sexp}_\eta(z) \), since \( \text{sexp}_\eta(0)=1 \), \( \text{sexp}_\eta(-1)=0 \), and there is a singularity at \( \text{sexp}_\eta(-2) \). \( \text{sexp}_\eta(z) \) does not grow superexponentially, but converges towards e as z increases. I recently posted the Taylor series for that function, centered at z=0, here:
http://math.eretrandre.org/tetrationforu...17#pid5817
- Sheldon
This is more what I was looking for, thanks. And another question, do you have a similar taylor series for \( \text{slog}_\eta(z) \)? That would be the inverse of the lower super function.
(06/02/2011, 10:27 PM)sheldonison Wrote: Jay suggested defining cheta(0)=2e. Then \( \text{cheta}(1)=\eta^{2e}=e^2 \), and cheta(2)=e^e, which seems like a reasonable choice for how to define cheta(0).
Woah! have you ever thought to consider that since \( \text{cheta}(0) = 2*e = e + e \), \( \text{cheta}(1) = e^2 = e*e \), and \( \text{cheta}(2) = \text{sexp}_e(2) = e^e \) that the cheta function maps the growth of \( e\, \{x\}\, e \), where \( \{x\} \) is a hyper operator of x order (0 is addition, 1 is multiplication etc etc); or mathematically speaking, another conjecture:
\( \text{cheta}(x)\, =\, e\, \{x\}\, e\, =\, e\,\{x+1\}\,2 \), which should at least be true over domain [0, 2]. To see if its universally true would be very difficult, though.
