Discussion of References

[1] Robbins A, Solving for the Analytic Piecewise Extension of Tetration and the Super-logarithm, 2005, Available from: (https://andydude.github.io/tetration/archives/tetration2/pdf/TetrationSuperlog_Robbins.pdf)

[2] Hollen M, Attempting to Define Tetration of Non-Integer Heights, 2021, Available from:https://symposium.foragerone.com/2022-csef/presentations/38304

[3] Ueda T, Extension of tetration to real and complex heights, 2021, Available from: https://www.academia.edu/67569880/Extension_of_tetration_to_real_and_complex_heights

[4] Maciuk A, Remarks About the Square Equation, Functional Square Root of a Logarithm, 2016, Available from: https://www.researchgate.net/publication/327169196_REMARKS_ABOUT_THE_SQUARE_EQUATION_FUNCTIONAL_SQUARE_ROOT_OF_A_LOGARITHM, 

[5] Almismari N, The first derivative proof of b^^x, x^^x, and f^^f(x) by differentation fundamental limits method, 2013,  Available from: https://vixra.org/pdf/1305.0040v1.pdf

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As stated in the text of the paper, the author considers efforts to estimate tetration with

non-integer heights using Taylor series to be unfruitful approach. The derivative of

      x

y =   e is an infinite multiplicative series which results in complex values (and infinite values

​

when x is an integer). It is difficult to see how a Taylor series can be used when just the first

 

derivative is problematic.