Pictures of some generalized analytical continuations

[Caleb]

I'm hesitant to post these pictures, as people might not understand. We are taking \(z^{16} =-1\) sample points. And we are writing:

\[
H(z) = \sum_{n=0}^\infty \frac{z^n}{\left(1+z^n\right)^n} \frac{1}{2^n}\\
\]

And,

\[
G(z) = \sum_{n=0}^\infty \frac{z^n}{\left(1+z^n\right)^{n^2}} \frac{1}{2^n}\\
\]

I opted for higher res pictures of these objects; just because they encompass more chaos. But they are graphed over the same box domain: \(|\Re(z)| < 2\) and \(|\Im(z)|<2\). 

Here is \(H\):



Here is \(G\):

---

These functions have a reflection formula... And that's my main point....

I'm going to take \(z^{40} = -1\) amount of sample points; and make a much bigger graph, with even higher res; but just for \(H(z)\). This will take 5-6 hours at best...

In the mean time, here is \(z^{40} =-1\) and a hi res graph of Caleb's original function...

What function are you refering to as "Caleb's" function? Is this
\[
G(z) = \sum_{n=0}^\infty \frac{z^n}{\left(1+z^n\right)} \frac{1}{2^n}\\
\]
When I graph this function I get a bit more constrast between the inner and outer function. For instance, if I plot the series of the first 200 terms in Mathemtica, I get this graph

Which definitely seems to have that kind of unnatural jump. I see this same thing when I graph the first 100 terms in my plotter

In fact, here is a plot of the real part of first 1000 terms at the angle \( \theta = \frac{\pi}{5.5} \) (i.e. I'm graphing \( f(r*e^{i \theta}) \) from r = -2 to 2)

And you can kind of notice the unnatural increasing on the right side. Something similarly weird happens at other angles, for instance at \( \theta = \frac{\sqrt{2}}{2} \pi \) we get the graph 

You can see the weirdness if we plot the derivative of the function above. You get an odd and sudden switch in the behaviour of the function near 1. For instance, a plot of the derivative of the first 2000 terms of the sum at the angle  \( \theta = \frac{\sqrt{2}}{2} \pi \) gives the following graph

This of course doesn't mean anything definitive, but it suggests that even matching all derivatives on the boundary might not fully resolve the issue of finding the right continuation.