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.