Seeing Ember Edison, and Leo talk about cases which cannot be solved using a theta mapping--I thought I'd see if the infinite composition manner is feasible with these anomalous values. I'm only going to sketch an approach here, and try to construct a real valued tetration \( H_\lambda(s) \) where:
\(
H_\lambda(s)\,\,\text{is real valued for}\,\,\lambda \in \mathbb{R}^+\\
H_\lambda(0) = 1\\
2^{-H_\lambda(s)} = H_\lambda(s+1)\\
H_\lambda(s+2\pi i / \lambda) = H_\lambda(s)\\
H\,\,\text{is holomorphic for}\,\,0 < |\Im(s)| < \pi/\lambda\,\,\text{for}\,\,\lambda \in \mathbb{R}^+\\
\)
To begin, we find a function that approximates this tetration. Let's call this function \( \varphi(s) \) which is holomorphic on \( \mathbb{C}/\{\lambda(j-s) = (2k+1) \pi i\,\,j \ge 1,\,k \in \mathbb{Z}\} \) and has a period of \( 2 \pi i / \lambda \). This function will satisfy the functional equation:
\(
\varphi_\lambda(s+1) = \frac{2^{-\varphi_\lambda(s)}}{e^{-\lambda s} + 1}\\
\)
These can be expressed as,
\(
\varphi_\lambda(s) = \Omega_{j=1}^\infty \frac{2^{-z}}{e^{\lambda(j-s)}+1}\,\bullet z\\
\)
Where if \( q_j(s,z)=\frac{2^{-z}}{e^{\lambda(j-s)}+1} \); then this expression equals:
\(
\varphi_\lambda(s)=\lim_{n\to\infty}q_1(s,q_2(s,...q_n(s,z)))\\
\)
Which converges compactly uniformly on the above domain because the sum,
\(
\sum_{j=1}^\infty ||\frac{2^{-z}}{e^{\lambda(j-s)}+1}||_{\mathcal{K},\mathcal{U}} < \infty\\
\)
converges compactly uniformly for \( z \in \mathcal{K} \subset \mathbb{C} \) and s in a compact set \( \mathcal{U} \) of the above domain.
Now, what we want to do is insert an error term \( \mu_\lambda \) such that,
\(
H_\lambda(s+s_0) = \varphi_\lambda(s) + \mu_\lambda(s)\\
\)
for some normalization constant \( s_0 \). To define the error term \( \mu_\lambda \) we use a sequence of functions:
\(
\mu_\lambda^{n+1}(s) = \log(1+e^{-\lambda s})/\log(2) -\log(1+\frac{\mu_\lambda^n(s+1)}{\varphi_\lambda(s+1)})/\log(2)\\
\)
Now, I'm not going to show this converges, but adapting the case from \( e \)--this shouldn't be hard to show converges. Instead, I'll just post some graphs showing the structure. These are only accurate to about 9 digits, so far. I just wrote together a quick script.
Here's \( \varphi_1 \)--so this has a 2 pi I period; and is real valued. This is over \( x \in [-2,2] \)
Here's our function \( \varphi_1(x) + \mu^{100}_1(x) \) over \( x \in [-1.5,4] \):
And here's our function \( [\Re(\varphi_1(x+i) + \mu^{100}_1(x+i)), \Im(\varphi_1(x+i) + \mu^{100}_1(x+i))] \) over \( x \in [-1.5,4] \)
Again, both of these graphs satisfy \( 2^{-H_1(s)} = H_1(s+1) \) to about 9 digits.
So all in all, I'm very confident that the infinite composition method will work for \( b = 1/2 \). I'm wondering if a similar result will hold for complex bases; but I'm not there yet. I'll try to write a script for that in a bit; for the moment I thought I'd just post the function \( H_1(s) \).
As I'm studying this more, we do not get \( H_1 \) as I thought we do. Branch cuts appear flippantly in \( 0<\Im(s) < \pi \); which I wasn't expecting, but still isn't very unexpected. I'm currently compiling a graph of \( \varphi_1(s) + \mu_1^{100}(s) \) in the complex plane. And it's pretty wonky so far. But is definitely locally holomorphic almost everywhere. My proof that no branch cuts occur in \( b = e \) do not carry over to \( 0<b < 1 \) so it's reasonable to see branch cuts. The negative logs throw a good wrench in the gears.
We also can run an indefinite amount of iterations; which is really nice. We can't do this with \( b=e \) without hitting overflow errors. I'll post this graph tomorrow when it finishes compiling.
\(
H_\lambda(s)\,\,\text{is real valued for}\,\,\lambda \in \mathbb{R}^+\\
H_\lambda(0) = 1\\
2^{-H_\lambda(s)} = H_\lambda(s+1)\\
H_\lambda(s+2\pi i / \lambda) = H_\lambda(s)\\
H\,\,\text{is holomorphic for}\,\,0 < |\Im(s)| < \pi/\lambda\,\,\text{for}\,\,\lambda \in \mathbb{R}^+\\
\)
To begin, we find a function that approximates this tetration. Let's call this function \( \varphi(s) \) which is holomorphic on \( \mathbb{C}/\{\lambda(j-s) = (2k+1) \pi i\,\,j \ge 1,\,k \in \mathbb{Z}\} \) and has a period of \( 2 \pi i / \lambda \). This function will satisfy the functional equation:
\(
\varphi_\lambda(s+1) = \frac{2^{-\varphi_\lambda(s)}}{e^{-\lambda s} + 1}\\
\)
These can be expressed as,
\(
\varphi_\lambda(s) = \Omega_{j=1}^\infty \frac{2^{-z}}{e^{\lambda(j-s)}+1}\,\bullet z\\
\)
Where if \( q_j(s,z)=\frac{2^{-z}}{e^{\lambda(j-s)}+1} \); then this expression equals:
\(
\varphi_\lambda(s)=\lim_{n\to\infty}q_1(s,q_2(s,...q_n(s,z)))\\
\)
Which converges compactly uniformly on the above domain because the sum,
\(
\sum_{j=1}^\infty ||\frac{2^{-z}}{e^{\lambda(j-s)}+1}||_{\mathcal{K},\mathcal{U}} < \infty\\
\)
converges compactly uniformly for \( z \in \mathcal{K} \subset \mathbb{C} \) and s in a compact set \( \mathcal{U} \) of the above domain.
Now, what we want to do is insert an error term \( \mu_\lambda \) such that,
\(
H_\lambda(s+s_0) = \varphi_\lambda(s) + \mu_\lambda(s)\\
\)
for some normalization constant \( s_0 \). To define the error term \( \mu_\lambda \) we use a sequence of functions:
\(
\mu_\lambda^{n+1}(s) = \log(1+e^{-\lambda s})/\log(2) -\log(1+\frac{\mu_\lambda^n(s+1)}{\varphi_\lambda(s+1)})/\log(2)\\
\)
Now, I'm not going to show this converges, but adapting the case from \( e \)--this shouldn't be hard to show converges. Instead, I'll just post some graphs showing the structure. These are only accurate to about 9 digits, so far. I just wrote together a quick script.
Here's \( \varphi_1 \)--so this has a 2 pi I period; and is real valued. This is over \( x \in [-2,2] \)
Here's our function \( \varphi_1(x) + \mu^{100}_1(x) \) over \( x \in [-1.5,4] \):
And here's our function \( [\Re(\varphi_1(x+i) + \mu^{100}_1(x+i)), \Im(\varphi_1(x+i) + \mu^{100}_1(x+i))] \) over \( x \in [-1.5,4] \)
Again, both of these graphs satisfy \( 2^{-H_1(s)} = H_1(s+1) \) to about 9 digits.
So all in all, I'm very confident that the infinite composition method will work for \( b = 1/2 \). I'm wondering if a similar result will hold for complex bases; but I'm not there yet. I'll try to write a script for that in a bit; for the moment I thought I'd just post the function \( H_1(s) \).
As I'm studying this more, we do not get \( H_1 \) as I thought we do. Branch cuts appear flippantly in \( 0<\Im(s) < \pi \); which I wasn't expecting, but still isn't very unexpected. I'm currently compiling a graph of \( \varphi_1(s) + \mu_1^{100}(s) \) in the complex plane. And it's pretty wonky so far. But is definitely locally holomorphic almost everywhere. My proof that no branch cuts occur in \( b = e \) do not carry over to \( 0<b < 1 \) so it's reasonable to see branch cuts. The negative logs throw a good wrench in the gears.
We also can run an indefinite amount of iterations; which is really nice. We can't do this with \( b=e \) without hitting overflow errors. I'll post this graph tomorrow when it finishes compiling.
