That's even more enlightening! That's a great intuitive way of interpreting it. That's the best way I can interpret it so far, I just get the intuition of the matter. I definitely couldn't reproduce the proof, or teach a class on the proof, but I'm really starting to get how it works. It's so inventive it's mindblowing.
The more I think about how Kneser does the impossible with this, the more I imagine there's some ridiculously clever way of producing \( ^ze \) from \( ^z\eta \). Much like the base change formula, but I imagine it would have to be less direct. I have this sneaking hunch that \( ^z\eta \) is the key to a nice tetration for all bases, or at least the real positive line for \( b>1 \). The bounded case is just too simple and unbelievably well behaved, and it has to be a gateway, or must shed some light on the unbounded case. Especially \( \eta \) because \( ^z\eta \) is holomorphic on the exact same domain that a decent \( ^ze \) extension would be holomorphic. Namely \( \mathbb{C}/\{(-\infty,-2)\} \). There must be some sequence of steps which produce \( ^ze \) from \( ^z\eta \). That's why I even study the bounded hyper-operators, it has to produce the unbounded case in some ingenial manner. At least, that's the goal, lol.
Thanks a lot for these descriptions, Sheldon. That's why I've come here since highschool. Such a nice and "no question is a bad question" forum. I can't believe how much more rigorous and well-founded my arguments about hyper-operators have become since I've come here. Too bad it's been less active lately.
Plus it's unreal that that tiny region becomes the half plane and it preserves the composition by \( e^z \) and it solves tetration. Who on god's earth could come up with that?
The more I think about how Kneser does the impossible with this, the more I imagine there's some ridiculously clever way of producing \( ^ze \) from \( ^z\eta \). Much like the base change formula, but I imagine it would have to be less direct. I have this sneaking hunch that \( ^z\eta \) is the key to a nice tetration for all bases, or at least the real positive line for \( b>1 \). The bounded case is just too simple and unbelievably well behaved, and it has to be a gateway, or must shed some light on the unbounded case. Especially \( \eta \) because \( ^z\eta \) is holomorphic on the exact same domain that a decent \( ^ze \) extension would be holomorphic. Namely \( \mathbb{C}/\{(-\infty,-2)\} \). There must be some sequence of steps which produce \( ^ze \) from \( ^z\eta \). That's why I even study the bounded hyper-operators, it has to produce the unbounded case in some ingenial manner. At least, that's the goal, lol.
Thanks a lot for these descriptions, Sheldon. That's why I've come here since highschool. Such a nice and "no question is a bad question" forum. I can't believe how much more rigorous and well-founded my arguments about hyper-operators have become since I've come here. Too bad it's been less active lately.
Plus it's unreal that that tiny region becomes the half plane and it preserves the composition by \( e^z \) and it solves tetration. Who on god's earth could come up with that?