The periodic manner makes a bit more sense, but I'm confused how we get \( ^{-1} e = 0 \) since the original super function never equals zero. Is this just playing tricks with an essential singularity making a zero pop out? But then, how is it holomorphic in a neighborhood of zero? There must be some trick I'm missing.
I guess what I don't like about Kneser's solution is that there is no way to apply the same techniques to get from tetration to pentation. We inherently use that \( e^z \) is entire, so the same idea won't work for \( ^z e \) since it isn't entire. For example, it won't have an entire inverse Schroder function.
I think I'm more prone to Kouznetsov's method, since it appears we can get to pentation from it, and so on and so forth. Sadly the fact Kouznetsov wasn't able to prove the representation actually converges is a real downer.
I think an important facet of solving for tetration is that the same technique can be used to solve for pentation, then hexation, so on and so forth. That's probably why I just love the bounded case so much! To get from tetration to pentation we literally just do what we did to get from exponentiation to tetration.
I guess what I don't like about Kneser's solution is that there is no way to apply the same techniques to get from tetration to pentation. We inherently use that \( e^z \) is entire, so the same idea won't work for \( ^z e \) since it isn't entire. For example, it won't have an entire inverse Schroder function.
I think I'm more prone to Kouznetsov's method, since it appears we can get to pentation from it, and so on and so forth. Sadly the fact Kouznetsov wasn't able to prove the representation actually converges is a real downer.
I think an important facet of solving for tetration is that the same technique can be used to solve for pentation, then hexation, so on and so forth. That's probably why I just love the bounded case so much! To get from tetration to pentation we literally just do what we did to get from exponentiation to tetration.