Intervals of Tetration

[bo198214]

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.

Though it contains a heap of errors:

• parabolic iteration works merely for the case \( b=e^{1/e} \). Jabotinsky's double binomial expansion works only in the parabolic case.
  
• Helm's method works generally for \( b>1 \) and yields real values. For \( 0<b<1 \) it obtains complex values.
  
• Woon's series was not designed for functions but for operators. I think it is equal to the parabolic case for \( w=1 \) and using a function instead of the operator.
  
• I think Jay's method works merely for \( b>e^{1/e} \).
  
• I think there are two solutions by Ioannis, one that is merely continuous but satisfies \( {}^{x}b = b^{\left({}^{(x-1)}b\right)} \) and one that is \( C^\infty \) and satisfies the condition only for integer \( x \).