tetration exp(z)-1+k

[tommy1729]

[quote='sheldonison' pid='7609' dateline='1422314910']
where k is a small perturbation constant in the neighborhood of zero...


First off, if \( k=\ln(ln(b))+1\;\; \) then \( s_k(z) = \ln(b)\cdot \text{sexp}\left(\text{slog}(e)-2\right)\;+\;\ln\left(\ln(b)\right)\; \) with some algebra

Despite the edit this equation is still wrong.
Look , your "z" is on the LHS but not on the RHS.

\( \frac{d}{dz}s_k(z) = \frac{d}{dz}s_k(z+1)\;\; \) trivially proven by using the chain rule since \( s_k(z+1)=\exp(s_k(z))-1+k \)

This is also wrong.
sexp'(x+1) = sexp(x+1) sexp'(x).

If you say f ' (x) = f ' (x+1) then basicly you say f ' (x) is periodic.

So, this particular point on the tetration curve is the place where the linear approximation piecemeal approximation works best, since
...

Considering these mistakes or miscommunications I have trouble following and as a reader am not seduced to do so.
Maybe Im harsch but if you ever want to write a paper you will loose the intrest of people after page 1 by these things , if it even gets accepted.

As for my opinion , well , considering what I wrote its clear.
Also the idea or post seems unfinished and is therefore hard to Judge.
Nothing new under the sun till now ?

regards

tommy1729