Intervals of Tetration

[jaydfox]

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).
  

Don't forget Peter Walker's solution in his paper Infinitely Differentiable Generalized Logarithmic and Exponential Functions.

I've gone back and played with his solution, and I think it gives the same results as mine, at least for converting from base eta to base e. My change of base formula is essentially the same as his "h" function (though his "h" function is specific for converting between base e and base eta, using a double-logarithmic scale), and his "g" function is a painfully slow way of calculating the parabolic continuous iteration of e^z-1, which is the equivalent of the double-logarithm of my cheta function.

My change of base formula requires a proper superexponential function, which my cheta function is. The parabolic continuous iteration of the decremented natural logarithm is not. However, I work in double-logarithmic math to use the change of base formula, and the double logarithm of my cheta function is the parabolic iteration of e^z-1. So both our solutions use the same underlying math, but calculate the results in different ways.

I've already noted that my solution and Andrew's are different, and Peter himself noted that his (h/g) solution appears to be different from some "matrix method", which is likely similar to or the same as Andrew's slog.