02/14/2016, 07:55 AM
(This post was last modified: 02/14/2016, 04:14 PM by sheldonison.)
sheldon Wrote:See http://math.eretrandre.org/tetrationforu...729&page=2 post#15,
In that post, there is a Taylor series for the 1st derivative of sexp_b(z) developed around b=2, and a Taylor series for sexp_b(-0.5), also developed around b=2. At the time I posted that, I also had developed Taylor series for the first 50 or so derivatives in the neighborhood of b=2.
(02/14/2016, 05:08 AM)marraco Wrote: As I understand, those are series for the variable in the exponent, but we need series for the variable in the base; a series for sexp_{x+r}(1.5). I mean not the series for \( \\[15pt]
{^xb} \), but for \( \\[15pt]
{^{1.5}(x+r)} \). (I write r instead of 1).
That's why I used your knesser.gp to calculate points for different bases separated by n*dx steps...
Those series are for the base not the exponent! That's why they needed to be generated with tetcomplex rather than kneser; because getting such amazingly good analytic convergence for such a series requires complex base tetration, with results for n bases, equally spaced around a circle centered on base=2. fatou.gp (with small code updates) could also generate such results. To evaluate those two referenced Taylor series, substitute x=(b-2), to evaluate the Taylor series.
This is probably off topic, but bases between 1 and eta=exp(1/e) aren't the same function as Kneser tetration for real bases>eta. Also, when you develop the iterated exponential using regular iteration from the attracting fixed point for bases<eta, there is no super-exponential growth anywhere in the complex plane. There is a tetration function for bases between 1 and eta, but it is no longer real valued at the real axis. Regular iteration for bases<eta is a much simpler problem than tetration, but unfortunately it cannot be extended analytically in the complex plane to tetration for real bases>eta because it is a completely different function than Kneser tetration.
- Sheldon
