Oh, and I forgot to mention some other connections:
First, the logit is also approachable as \[{\rm logit}[f]=\frac{\partial f^{\circ t}(x)}{\partial t}\big|_{t=0}\]
In our case we know that the regular iterations are \(f^{\circ t}(x)=-{\rm arccot}(t+\cot(x))\) hence
\[ j = {\rm logit}[f] = \frac{\partial f^{\circ t}(x)}{\partial t}\big|_{t=0}= \frac{1}{(t - \cot(x))^2 + 1}\big|_{t=0} = \frac{1}{\cot(x)^2+1} \]
Also the julia function/logit can be used to reconstruct the Abel function which is \(-\cot(x)\) in our case:
\[ \alpha(x) = \int \frac{dx}{j(x)} = \int (\cot(x)^2+1) dx = -\cot(x)\]
From the standpoint of formal powerseries this is even more interesting:
\[ \alpha(x) = \int \frac{dx}{j(x)} = \int x^{-2} + \frac{1}{3} + \frac{1}{15}x^2 + \frac{2}{189}x^4 + \frac{1}{675}x^6 + \frac{2}{10395}x^8 + ... dx \]
Interesting here is that it does not contain the \(x^{-1}\) term, which in turn when we take the integral does not result in a \(\log\) term, \(\cot\) only has a pole:
\[ \alpha(x) = - x^{-1} + \frac{1}{3}x + \frac{1}{45}x^3 + \frac{2}{945}x^5 +\frac{1}{4725}x^7 + \frac{2}{93555}x^9 + ...\]
And this pole (instead of a log-involved singularity) was the warranty for the smoothness of \(-{\rm arccot}(t-\cot(x))\).
First, the logit is also approachable as \[{\rm logit}[f]=\frac{\partial f^{\circ t}(x)}{\partial t}\big|_{t=0}\]
In our case we know that the regular iterations are \(f^{\circ t}(x)=-{\rm arccot}(t+\cot(x))\) hence
\[ j = {\rm logit}[f] = \frac{\partial f^{\circ t}(x)}{\partial t}\big|_{t=0}= \frac{1}{(t - \cot(x))^2 + 1}\big|_{t=0} = \frac{1}{\cot(x)^2+1} \]
Also the julia function/logit can be used to reconstruct the Abel function which is \(-\cot(x)\) in our case:
\[ \alpha(x) = \int \frac{dx}{j(x)} = \int (\cot(x)^2+1) dx = -\cot(x)\]
From the standpoint of formal powerseries this is even more interesting:
\[ \alpha(x) = \int \frac{dx}{j(x)} = \int x^{-2} + \frac{1}{3} + \frac{1}{15}x^2 + \frac{2}{189}x^4 + \frac{1}{675}x^6 + \frac{2}{10395}x^8 + ... dx \]
Interesting here is that it does not contain the \(x^{-1}\) term, which in turn when we take the integral does not result in a \(\log\) term, \(\cot\) only has a pole:
\[ \alpha(x) = - x^{-1} + \frac{1}{3}x + \frac{1}{45}x^3 + \frac{2}{945}x^5 +\frac{1}{4725}x^7 + \frac{2}{93555}x^9 + ...\]
And this pole (instead of a log-involved singularity) was the warranty for the smoothness of \(-{\rm arccot}(t-\cot(x))\).
