02/19/2013, 05:10 PM
(This post was last modified: 02/19/2013, 08:48 PM by sheldonison.)
(02/19/2013, 11:18 AM)Balarka Sen Wrote: ... I recently read the kneser's solution to the tetration and saw a code here, in the forum, somewhere. But my problem is that the code seems to be initializing the base and prints all the information on the PARI screen. I want to simultaneously calculate sexp(b, h) for different bases but same heights which is a bit difficult for me using that code, can anyone give me a code that will initialize the height first and can give results for different bases at once?
Hi Balarka. I'm Sheldon, the author of the pari-gp code that you probably downloaded. The algorithm has to calculate the equivalent of Kneser's Riemann mapping for each base, before it can generate results, so it won't give you results for different bases as easily as you might like.
Quote:However, I have a question regarding knesers solution to the extended tetration : is it being proved that knesner's method converges for any complex height? I searched a bit (and tried with a pen and paper too) but haven't found anything regarding it.
Yes, Kneser's solution for real bases is proven analytic in the upper/lower halves of the complex plane, with singularities at the real axis for negative integers<=-2. I did post a Taylor series for \( \text{sexp}_a (-0.5) \) centered at base a=2. Look for "Taylor series for sexp(-0.5)" in the thread with the complex base pari-gp code, http://math.eretrandre.org/tetrationforu...729&page=2
The pari-gp code I wrote that extends tetration to complex bases is much more limited in how well it converges. I regarded the code as an effort to stimulate the discussion as to how to extend tetration to complex bases, based on Mike's ideas. Below is the Taylor series for \( ^{-0.5} a \), for complex tetration base a centered at a=2. Convergence is limited by the nearest singularity at base \( \eta=\exp(\frac{1}{e})\approx1.444667861 \), though that is a mild singularity, and using fewer Taylor series terms gives convergence over a wider range, limited more by the singularity at base=1.
- Sheldon
Code:
{ sexp_mhalf = /* sexp(-0.5) for base a-2 */
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}