02/02/2021, 08:37 AM
Hey, everyone!
Hope everyone is doing well, and doing math, and having fun. I'm having loads of fun working on tetration again. So I added twenty or so pages to my last paper. Now, the major difference in this iteration of the paper is actually something really small. But once you see this small error, it collapses my entire approach in the first iteration.
In my original iteration, the ten or so pages, I assumed that,
\(
\psi_{\pi}(t) \le |\psi_y(t)|\\
\)
Everywhere! Which turns out to be wrong. So I spent a whole lot of time, after talking to Sheldon, on clarifying everything I know, drawing out every string--and consequently showing more or less the same result. But now it's,
\(
\forall \delta>0\exists T \forall t\ge T\,\,\psi_{\pi}(t) \le |\psi_y(t)|\,\,\text{if}\,\,\delta < y < 2\pi - \delta\\
\)
It's a curse and a blessing, because it took 20 more pages; I had to rewrite a good amount of the proofs; taught me where I should better explain; reminded me to pull on every thread; and so on... I still see a couple holes in the dam that my finger has to plug for the moment. But I'm so close; I can smell it.
So I present a 29 page paper roughly constructing a tetration function \( e \uparrow \uparrow s \) that is holomorphic on \( \mathbb{C}/(\mathbb{R} + 2\pi ik) \) and at least continuously differentiable on \( \mathbb{R} + 2\pi i k \) excluding singularities.
Hope everyone is doing well, and doing math, and having fun. I'm having loads of fun working on tetration again. So I added twenty or so pages to my last paper. Now, the major difference in this iteration of the paper is actually something really small. But once you see this small error, it collapses my entire approach in the first iteration.
In my original iteration, the ten or so pages, I assumed that,
\(
\psi_{\pi}(t) \le |\psi_y(t)|\\
\)
Everywhere! Which turns out to be wrong. So I spent a whole lot of time, after talking to Sheldon, on clarifying everything I know, drawing out every string--and consequently showing more or less the same result. But now it's,
\(
\forall \delta>0\exists T \forall t\ge T\,\,\psi_{\pi}(t) \le |\psi_y(t)|\,\,\text{if}\,\,\delta < y < 2\pi - \delta\\
\)
It's a curse and a blessing, because it took 20 more pages; I had to rewrite a good amount of the proofs; taught me where I should better explain; reminded me to pull on every thread; and so on... I still see a couple holes in the dam that my finger has to plug for the moment. But I'm so close; I can smell it.
So I present a 29 page paper roughly constructing a tetration function \( e \uparrow \uparrow s \) that is holomorphic on \( \mathbb{C}/(\mathbb{R} + 2\pi ik) \) and at least continuously differentiable on \( \mathbb{R} + 2\pi i k \) excluding singularities.