A question about tetration from a newbie

[TetrationSheep]

Hello, I am pretty new to tetration, so sorry if this question is a little bit simple, but how can we define a specific mapping between the inverse Abel function, and Kneser's construction of tetration? Specifically, how could we map the Abel function to the unit circle?

[bo198214]

Hello, I am pretty new to tetration, so sorry if this question is a little bit simple, but how can we define a specific mapping between the inverse Abel function, and Kneser's construction of tetration? Specifically, how could we map the Abel function to the unit circle?

Yeah, that's the big computational question! The existence of a mapping from the Abel function to the unit circle is guarantied by the Riemann mapping theorem, however if you follow the proof of the theorem for finding a numerical algorithm, then it's quite not suited for numerical handling. That's why there are so many different algorithms on this board, to compute the Abel function or it's inverse and not all might be equal to Kneser's solution. Actually for most of the numerical algorithms here there is not even a proof of convergence, not to talk about equaltiy of different algorithms!

[TetrationSheep]

Hello, I am pretty new to tetration, so sorry if this question is a little bit simple, but how can we define a specific mapping between the inverse Abel function, and Kneser's construction of tetration? Specifically, how could we map the Abel function to the unit circle?

Yeah, that's the big computational question! The existence of a mapping from the Abel function to the unit circle is guarantied by the Riemann mapping theorem, however if you follow the proof of the theorem for finding a numerical algorithm, then it's quite not suited for numerical handling. That's why there are so many different algorithms on this board, to compute the Abel function or it's inverse and not all might be equal to Kneser's solution. Actually for most of the numerical algorithms here there is not even a proof of convergence, not to talk about equaltiy of different algorithms!

Oh, so that's why there's no consensus! I thought these methods were just competing for convergence/accuracy. Thanks for informing me about this!