Intervals of Tetration

[Daniel]

Methods of Tetration

• Koch/Bell/Carleman matrices (includes parabolic and hyperbolic iteration) -- works for \( b \in [e^{-e}, e^{1/e}] \) (although it works outside this range, it produces complex values).
  
  • Daniel Geisler's parabolic tetration series -- first 3 derivatives of parabolic iteration.
    
  • Daniel Geisler's hyperbolic tetration series -- first 3 derivatives of hyperbolic iteration.
    
  • Eri Jabotinsky's double-binomial expansion -- a simplification of parabolic iteration.
    
  • Helms' exp(t log(M)) method -- should work in the same interval.
    
• Iteration-based solution of Abel FE (Peter Walker's)-- only given for \( b = e \).
  
• Matrix-based solution of Abel FE (Andrew Robbins')-- works for \( b > 1 \) (although it converges faster for \( b > e^{1/e} \)).
  
• S.C.Woon's series (w=1) -- quickly converges for \( b \in [0, 1] \) (but straight line), may converge for \( b \in [1, e^{1/e}] \), diverges for \( b > e^{1/e} \).
  
• Ioannis Galidakis' solution -- does not satisfy \( {}^{x}b = b^{\left({}^{(x-1)}b\right)} \) for all x (only for integer x).
  
• Cliff Nelson's hyper-logarithms -- not continuous, but defines all hyper-(n+1)-logarithms in terms of hyper-n-logarithms.
  
• Robert Munafo's solution -- seems C^n, but defined by a nested exponential, so hard to determine analycity.
  
• Jay Fox's change-of-base -- theoretically speaking, should work for all \( b > 1 \).
  
• Ingolf Dahl's solution -- based on fractional iteration (interval unknown).
  

This is a nice list of different approaches to tetration. While I would like to extend my work to matrices, it currently doesn't involve matrices althought I did find that I obtained results experimentally that were consistent with Bell matrices. I use Faa di Bruno's formula for \( f(g(x)) \), set \( g(x) = f^{n-1}(x) \) and solve. My Mathematica software has computed the first 8 derivatives of \( f^{n}(x) \). This then simplifies for the cases of parabolic and hyperbolic iteration where I have only listed the first three derivatives for the sake of brevity.

Daniel Geisler