03/11/2023, 04:03 AM
I want to share some illustrations to give others some context about what generalized analytical continuation looks like.
First, let me mention my general philosophy in studying an object like generalized analytical continuation. At the moment, math research is only a hobby, so my main motivation in studying these objects is purely aesthetic. Thus, my main goal is really to find the "most beautiful" way to continue a function beyond its boundary. Thus, some of the analysis in the following post will be motivated by what type of options seem most mathemtically elegent.
This first function
\[ f(x)= \sum_{n=1}^\infty \frac{x^n}{1+x^n} \frac{1}{n^2}\]
and it is continued in the natural way (i.e. by plugging in)
![[Image: 1U2zvJkKcB-YHlet9JN8HRB2MzFLQRV54]](https://lh3.googleusercontent.com/d/1U2zvJkKcB-YHlet9JN8HRB2MzFLQRV54)
To me, this graph seems a bit incoherent between the two parts, but this is just a feeling.
Here is a graph of the natural continuation of
\[f(x) = \sum_{n=0}^\infty \frac{1}{2^n} \frac{1}{1+x^n}\]
I've graphed this one in a slighty different way to bring out the structure better. I think the natural continuation is perfect for this function-- its hard to tell whats inside the boundary and what's not.
![[Image: 1ZwjXfZN-_7hXeIQOUzbb9GkN42hhp9SQ]](https://lh3.googleusercontent.com/d/1ZwjXfZN-_7hXeIQOUzbb9GkN42hhp9SQ)
However, the continuation seems to get worse when \(2^n\) gets closer to \(1^n\). For instance, looking at
\[f(x) = \sum \frac{1}{1.5^n}{1}{1+x^n}\]
We get the graph below, which doesn't seem as nice.
![[Image: 1rU_HH-Yz_ugHjjUZQVfvJvc7P7zi5gL_]](https://lh3.googleusercontent.com/d/1rU_HH-Yz_ugHjjUZQVfvJvc7P7zi5gL_)
One idea I have been considering is that there are cases where you already know the location of all the poles. For instance, consider Mick's function
\[f(x) = \sum_{k=0}^\infty \left(\frac{1}{1-\frac{x^{k+1}}{2^{k-1}}}-1\right) \]
We know that the poles have location \(x = 2 * \sqrt[k+1]{\frac{1}{2^2}}\). For instance, on the upper half of the disc we get the poles
![[Image: 1NT-wNhCpOVxAReBRPaIIAFUt6WTlGkdZ]](https://lh3.googleusercontent.com/d/1NT-wNhCpOVxAReBRPaIIAFUt6WTlGkdZ)
However, if we changed 2 to something less than 1, then we would know the location of the poles "should be" \( \alpha \cdot \sqrt[k+1]{\frac{1}{\alpha^2}}\), however we wouldn't be able to sum the series. This makes gives a sanity check to make sure that certain continuation techniques work with what we expect.
Also, another interesting thing to note is that in all of these graphs the poles and zeroes seem to always be paired. In particular, there always seems to be a natural way to pair up a zero inside the boundary with one on the outside. I guess this probably follows from the 1/x thing, but I still find it interesting.
Anyway, I'm thinking I'll just use this thread to keep track of my drawing and make them available to others in case it happens to be useful.
First, let me mention my general philosophy in studying an object like generalized analytical continuation. At the moment, math research is only a hobby, so my main motivation in studying these objects is purely aesthetic. Thus, my main goal is really to find the "most beautiful" way to continue a function beyond its boundary. Thus, some of the analysis in the following post will be motivated by what type of options seem most mathemtically elegent.
This first function
\[ f(x)= \sum_{n=1}^\infty \frac{x^n}{1+x^n} \frac{1}{n^2}\]
and it is continued in the natural way (i.e. by plugging in)
To me, this graph seems a bit incoherent between the two parts, but this is just a feeling.
Here is a graph of the natural continuation of
\[f(x) = \sum_{n=0}^\infty \frac{1}{2^n} \frac{1}{1+x^n}\]
I've graphed this one in a slighty different way to bring out the structure better. I think the natural continuation is perfect for this function-- its hard to tell whats inside the boundary and what's not.
However, the continuation seems to get worse when \(2^n\) gets closer to \(1^n\). For instance, looking at
\[f(x) = \sum \frac{1}{1.5^n}{1}{1+x^n}\]
We get the graph below, which doesn't seem as nice.
One idea I have been considering is that there are cases where you already know the location of all the poles. For instance, consider Mick's function
\[f(x) = \sum_{k=0}^\infty \left(\frac{1}{1-\frac{x^{k+1}}{2^{k-1}}}-1\right) \]
We know that the poles have location \(x = 2 * \sqrt[k+1]{\frac{1}{2^2}}\). For instance, on the upper half of the disc we get the poles
However, if we changed 2 to something less than 1, then we would know the location of the poles "should be" \( \alpha \cdot \sqrt[k+1]{\frac{1}{\alpha^2}}\), however we wouldn't be able to sum the series. This makes gives a sanity check to make sure that certain continuation techniques work with what we expect.
Also, another interesting thing to note is that in all of these graphs the poles and zeroes seem to always be paired. In particular, there always seems to be a natural way to pair up a zero inside the boundary with one on the outside. I guess this probably follows from the 1/x thing, but I still find it interesting.
Anyway, I'm thinking I'll just use this thread to keep track of my drawing and make them available to others in case it happens to be useful.