Hmmm, I'm a little confused as to how this causes a singularity at the moment. But this could explain what I've been calling "fractal hairs" which appear for certain depths of iteration.
They seem to be located at about the pull back of this point.
Just so I understand you,
If \( \beta(z-1)+z = 0 \) then \( \tau(z-1) = \infty \).
I don't quite understand why that matters though... Could you elaborate further?
Because if \( \beta(z) \approx 0 \) and \( \beta(z+1) \approx 1 \); there really shouldn't be a problem in finding a,
\( \tau(z) \approx 0 \) and \( \tau(z+1) \approx 0 \);
\(
\tau(z) = -\log(1+\exp(-z)) + \log(1+ \frac{\tau(z+1)}{\beta(z+1)}) \approx 0\\
\)
I don't see how that causes a singularity. But at,
\(
\tau(z-1) =- \log(1+\exp(1-z)) + \log(1+\frac{\tau(z)}{\beta(z)})\\
\)
I can see it getting very large, but beta is non zero. This can only blow up if \( -\tau(z) = \beta(z) \); they're both small, so I see it in the realm of possibility. This would then equate to:
\(
\exp(\beta(z-1)) = \exp(-z)\\
(1+\exp(-z))*\beta(z) = \exp(-z)\\
\tau(z) = -\exp(-z)/(1+\exp(-z))\\
\)
So these would be our bad points; where these functions intersect. Interesting; this will definitely be helpful.
I will add something though; which is what I have technically written in the paper. I had only ever argued from evidence for holomorphy on \( 0 < \Im(z) < \pi \) (obviously, too confidently)--the actual theorem I proved, did not require this--nor did the construction of the final tetration function. Recall \( \mathbb{L} = \{(s,\lambda) \in \mathbb{C}^2\,|\,\Re\lambda > 0,\,\lambda(j-s) \neq (2k+1)\pi i,\,j,k\in\mathbb{Z},\,j\ge 1\} \).
Theorem 5.1 Tetration Existence Theorem:
For \( (s,\lambda) \in \mathcal{L} \subset \mathbb{L} \) there exists a holomorphic tetration function \( F_\lambda(s) \) such that,
\(
F_\lambda(s+1) = e^{F_\lambda(s)}\\
\)
Where \( \mathbb{L}/\mathcal{L} \) is a measure zero set in \( \mathbb{C}^2 \).
Where the measure zero statement is basically saying; there are branch cuts, and I have no idea where; but almost everywhere this converges. Which still seems to be the case if there are singularities.
The real question I'm wondering now, if a similar argument of yours would translate over to the actual beta tetration we really care about \( \text{tet}_\beta \). I think it would be a fair amount more difficult, considering we lose a lot of the convenient algebra you've used; yet something like it may pop up. Hmmm. I'm having trouble understanding how that would happen.
Could you explain to me though, how you've derived \( \beta(z-1) + z = 0 \Rightarrow \beta(z) = - \tau(z) \)? I still don't quite understand. Additionally the value you've given me seems to evaluate fine for larger iterations.
Great work, Sheldon!
Regards, James
They seem to be located at about the pull back of this point.
Just so I understand you,
If \( \beta(z-1)+z = 0 \) then \( \tau(z-1) = \infty \).
I don't quite understand why that matters though... Could you elaborate further?
Because if \( \beta(z) \approx 0 \) and \( \beta(z+1) \approx 1 \); there really shouldn't be a problem in finding a,
\( \tau(z) \approx 0 \) and \( \tau(z+1) \approx 0 \);
\(
\tau(z) = -\log(1+\exp(-z)) + \log(1+ \frac{\tau(z+1)}{\beta(z+1)}) \approx 0\\
\)
I don't see how that causes a singularity. But at,
\(
\tau(z-1) =- \log(1+\exp(1-z)) + \log(1+\frac{\tau(z)}{\beta(z)})\\
\)
I can see it getting very large, but beta is non zero. This can only blow up if \( -\tau(z) = \beta(z) \); they're both small, so I see it in the realm of possibility. This would then equate to:
\(
\exp(\beta(z-1)) = \exp(-z)\\
(1+\exp(-z))*\beta(z) = \exp(-z)\\
\tau(z) = -\exp(-z)/(1+\exp(-z))\\
\)
So these would be our bad points; where these functions intersect. Interesting; this will definitely be helpful.
I will add something though; which is what I have technically written in the paper. I had only ever argued from evidence for holomorphy on \( 0 < \Im(z) < \pi \) (obviously, too confidently)--the actual theorem I proved, did not require this--nor did the construction of the final tetration function. Recall \( \mathbb{L} = \{(s,\lambda) \in \mathbb{C}^2\,|\,\Re\lambda > 0,\,\lambda(j-s) \neq (2k+1)\pi i,\,j,k\in\mathbb{Z},\,j\ge 1\} \).
Theorem 5.1 Tetration Existence Theorem:
For \( (s,\lambda) \in \mathcal{L} \subset \mathbb{L} \) there exists a holomorphic tetration function \( F_\lambda(s) \) such that,
\(
F_\lambda(s+1) = e^{F_\lambda(s)}\\
\)
Where \( \mathbb{L}/\mathcal{L} \) is a measure zero set in \( \mathbb{C}^2 \).
Where the measure zero statement is basically saying; there are branch cuts, and I have no idea where; but almost everywhere this converges. Which still seems to be the case if there are singularities.
The real question I'm wondering now, if a similar argument of yours would translate over to the actual beta tetration we really care about \( \text{tet}_\beta \). I think it would be a fair amount more difficult, considering we lose a lot of the convenient algebra you've used; yet something like it may pop up. Hmmm. I'm having trouble understanding how that would happen.
Could you explain to me though, how you've derived \( \beta(z-1) + z = 0 \Rightarrow \beta(z) = - \tau(z) \)? I still don't quite understand. Additionally the value you've given me seems to evaluate fine for larger iterations.
Great work, Sheldon!
Regards, James