I'm going to write a modified version of MakeGraph by mike3

[JmsNxn]

Alright; so I've managed to code in the Schroder function to my satisfaction, and the inverse schroder function. I've always wanted to do this; as I've only done it inside the Shell-Thron region. I'm going to get it all working more detailed; but for the moment; here's a little preview of the chistar function:

   

You can see the singularity at \(0\) and \(1\); and they continue at each \(\exp^{\circ n}(0)\).

Here's the same function graphed parametrically:

   

You can kind of see the beginning of the star like shape that Sheldon saw.

Now, I'm pretty confused as to how you would like to overlay this over the beta method?

I've been fiddling and I've written:

\[
\chi(\beta_{1,1}(x))\\
\]

And this is again a parametric plot in the complex plane. We get a weird looking star like shape that is more prominent.

   

We can make a closer idea by taking:

\[
g_\beta(X) = \beta_{1,1}(\log(\chi(X))/L)
\]

Which satisfies:

\[
g_\beta(e^X) = \beta_{1,1}(\log(L \chi(X))/L) =\frac{ e^{g_\beta(X)}}{1+e^{-X\log(L \chi(X))/L}}\\
\]

So it's pretty close to Sheldon's \(\Psi^{-1}(\chi(X)) = X\).

   

But I'm still not sure precisely what you'd like to draw..?

I can try to draw anything you'd like; I'd just like some more context. Still not even sure what sheldon is trying to draw!