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 where:

To begin, we find a function that approximates this tetration. Let's call this function which is holomorphic on and has a period of . This function will satisfy the functional equation:

These can be expressed as,

Where if ; then this expression equals:

Which converges compactly uniformly on the above domain because the sum,

converges compactly uniformly for and s in a compact set of the above domain.

Now, what we want to do is insert an error term such that,

for some normalization constant . To define the error term we use a sequence of functions:

Now, I'm not going to show this converges, but adapting the case from --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 --so this has a 2 pi I period; and is real valued. This is over

Here's our function over :

And here's our function over

Again, both of these graphs satisfy to about 9 digits.

So all in all, I'm very confident that the infinite composition method will work for . 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 .

As I'm studying this more, we do not get as I thought we do. Branch cuts appear flippantly in ; which I wasn't expecting, but still isn't very unexpected. I'm currently compiling a graph of 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 do not carry over to 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 without hitting overflow errors. I'll post this graph tomorrow when it finishes compiling.

To begin, we find a function that approximates this tetration. Let's call this function which is holomorphic on and has a period of . This function will satisfy the functional equation:

These can be expressed as,

Where if ; then this expression equals:

Which converges compactly uniformly on the above domain because the sum,

converges compactly uniformly for and s in a compact set of the above domain.

Now, what we want to do is insert an error term such that,

for some normalization constant . To define the error term we use a sequence of functions:

Now, I'm not going to show this converges, but adapting the case from --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 --so this has a 2 pi I period; and is real valued. This is over

Here's our function over :

And here's our function over

Again, both of these graphs satisfy to about 9 digits.

So all in all, I'm very confident that the infinite composition method will work for . 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 .

As I'm studying this more, we do not get as I thought we do. Branch cuts appear flippantly in ; which I wasn't expecting, but still isn't very unexpected. I'm currently compiling a graph of 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 do not carry over to 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 without hitting overflow errors. I'll post this graph tomorrow when it finishes compiling.