Here are two graphs of:
\[
f_2(z) = \sum_{n=0}^\infty \frac{z^n}{\left(1+z^n\right)^2} \frac{1}{2^n}\\
\]
And:
\[
f_3(z) = \sum_{n=0}^\infty \frac{z^n}{\left(1+z^n\right)^3} \frac{1}{2^n}\\
\]
Over the box domain \(- 2 \le \Re(z) \le 2\) and \(-2 \le \Im(z) \le 2\). This graphing protocol is done fairly "low res", as I didn't want to spend a day compiling it. But this is still very accurate. Every singularity is mapped to a black pixel; but the area near a singularity is mapped well enough. I can make these graphs "higher res", it just takes more time.
These functions have a reflection formula about \(z \mapsto 1/z\).
Taking \(f_2(z)\) we get:
Please note, that this will be more accurate if you increase terms used. I only used about 16 terms, and 20 digits. This will fill out more with more of a boundary as you progress further.
Taking \(f_3(z)\) we get:
I took a fairly lower precision version of this picture, because I don't want these to compile all night. But it's still what it should look like.
Lastly, I am going to take Caleb's function, when \(m =1\). This is the same function as your first post. It is just compiled through my code; and graphed in a low res manner:
I want to point out, that all three of these functions have a reflection formula...
Ultimately these graphs are only using sample points along \(z^{16} = -1\), which is pretty low demanding. But it is very accurate as a calculator at this point--about 5 digit accuracy. So,you won't see much change in the graph if, say, we take \(z^{100} = -1\) as our sample points. It will look more "hi-res", but it'll be the same function.
This is a symptom of just how well behaved these Lambert/Caleb functions are....
\[
f_2(z) = \sum_{n=0}^\infty \frac{z^n}{\left(1+z^n\right)^2} \frac{1}{2^n}\\
\]
And:
\[
f_3(z) = \sum_{n=0}^\infty \frac{z^n}{\left(1+z^n\right)^3} \frac{1}{2^n}\\
\]
Over the box domain \(- 2 \le \Re(z) \le 2\) and \(-2 \le \Im(z) \le 2\). This graphing protocol is done fairly "low res", as I didn't want to spend a day compiling it. But this is still very accurate. Every singularity is mapped to a black pixel; but the area near a singularity is mapped well enough. I can make these graphs "higher res", it just takes more time.
These functions have a reflection formula about \(z \mapsto 1/z\).
Taking \(f_2(z)\) we get:
Please note, that this will be more accurate if you increase terms used. I only used about 16 terms, and 20 digits. This will fill out more with more of a boundary as you progress further.
Taking \(f_3(z)\) we get:
I took a fairly lower precision version of this picture, because I don't want these to compile all night. But it's still what it should look like.
Lastly, I am going to take Caleb's function, when \(m =1\). This is the same function as your first post. It is just compiled through my code; and graphed in a low res manner:
I want to point out, that all three of these functions have a reflection formula...
Ultimately these graphs are only using sample points along \(z^{16} = -1\), which is pretty low demanding. But it is very accurate as a calculator at this point--about 5 digit accuracy. So,you won't see much change in the graph if, say, we take \(z^{100} = -1\) as our sample points. It will look more "hi-res", but it'll be the same function.
This is a symptom of just how well behaved these Lambert/Caleb functions are....