08/01/2022, 10:56 PM
Lmao!
Sorry, Bo! It appears you and Kouznetsov refer to regular iteration differently! That's what's screwing up.
Correct me if I'm wrong, but when you refer to regular iteration about \(L\) for base \(b= e\), let's say, you are referring to the Taylor solution about \(z\approx L\); I.e: the iteration that looks like:
\[
f^{\circ t}(z) = L + \lambda^t(z-L) + O(z^2)\\
\]
This function is entire in \(t\), and there exists some \(t_0\) such that \(f^{t_0}(z_0) = 1\), thereby, we can make a tetration \(f^{t+t_0}(z_0)\) which is a superfunction to \(f(z) = \exp(z)\), but it is not real valued. \(f^{1/2 + t_0} \not \in \mathbb{R}\). I.e. it's a tetration function, but it's not real valued
Kouznetsov refers to the regular iteration about \(b = e\), as the Kneser iteration. This is based on his manner of constructing the function. Which I'm sure you're well aware of. His manner on the real line is real valued, and is what he always calls regular iteration.
I think I've been conflating regular iteration and Kouznetsov regular iteration.
I apologize again, English is my first language, but I'm terrible with vernacular. So in that sense, your english is better than mine, lol.
Woe is the trouble of standardizing terminology, lol.
I apologize, and if you're still confused, please please please explain what I've said to make you confused.
Regards.
Sorry, Bo! It appears you and Kouznetsov refer to regular iteration differently! That's what's screwing up.
Correct me if I'm wrong, but when you refer to regular iteration about \(L\) for base \(b= e\), let's say, you are referring to the Taylor solution about \(z\approx L\); I.e: the iteration that looks like:
\[
f^{\circ t}(z) = L + \lambda^t(z-L) + O(z^2)\\
\]
This function is entire in \(t\), and there exists some \(t_0\) such that \(f^{t_0}(z_0) = 1\), thereby, we can make a tetration \(f^{t+t_0}(z_0)\) which is a superfunction to \(f(z) = \exp(z)\), but it is not real valued. \(f^{1/2 + t_0} \not \in \mathbb{R}\). I.e. it's a tetration function, but it's not real valued
Kouznetsov refers to the regular iteration about \(b = e\), as the Kneser iteration. This is based on his manner of constructing the function. Which I'm sure you're well aware of. His manner on the real line is real valued, and is what he always calls regular iteration.
I think I've been conflating regular iteration and Kouznetsov regular iteration.
I apologize again, English is my first language, but I'm terrible with vernacular. So in that sense, your english is better than mine, lol.
Woe is the trouble of standardizing terminology, lol.
I apologize, and if you're still confused, please please please explain what I've said to make you confused.
Regards.
