03/14/2015, 10:37 PM
(This post was last modified: 03/14/2015, 10:38 PM by sheldonison.)
(03/12/2015, 11:10 PM)marraco Wrote: This paper (how do I insert a link here? the icons don't work)
Henryk Trappmann, Dmitrii Kouznetsov, Methods for real analytic tetration
https://bbuseruploads.s3.amazonaws.com/b...ain.pdf%22
That paper argues that Ttr(a,-1) = 0, and Ttr(a,-2) = ln(0) = -∞
, so it limits the domain of tetration exponents to (-2,∞)
But it does not account for multiple valued results for Ttr(a,-1), which give meaningful values for any negative integer (-∞,∞).
As I argued, Ttr(a,-1) = i.n.2.pi/ln(a)
So, there are infinite values, and for each n, Ttr(a,-2) also gives infinite vales: ∞², and Ttr(a,-3) haves ∞³ answers
The analytic tetration function referenced in the link has a logarithmic singularity at -2. Take any real number z which is greater than -2. Winding around the singularity counter clockwise, between -2 and -3, and then returning to z, you wind up with \( \text{Tet}_e(z)+2\pi i \). Winding around the singularity clockwise, between -2 and -3, and returning to z, then you have \( \text{Tet}_e(z)-2\pi i \). This is well known. To understand the behavior of \( \text{Tet}_e(z) \), one has to understand the analytic function and it's Riemann surface.
- Sheldon